Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{0}^{1}(1-x){{x}^{5}}dx$ $y=\int _{0}^{1}{{x}^{5}}-{{x}^{6}}dx$ $y=\left(...
$y=\int _{0}^{1}x{{(1-x)}^{5}}dx$
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Maths
Using properties of determinants prove that: $\left|\begin{array}{lll}\mathrm{x}-3 & \mathrm{x}-4 & \mathrm{x}-\alpha \\ \mathrm{x}-2 & \mathrm{x}-3 & \mathrm{x}-\beta \\ \mathrm{x}-1 & \mathrm{x}-2 & \mathrm{x}-\gamma\end{array}\right|=0$, where $\alpha, \beta, \mathrm{y}$ are in AP.
Solution: Given that $\alpha, \beta, \gamma$ are in an $A P$, which means $2 \beta=\alpha+\gamma$ Operating $R_{3} \rightarrow R_{3}-2 R_{2}+R_{1}$ $\begin{array}{l} =\left|\begin{array}{ccc} x-3...
Using properties of determinants prove that: $\left|\begin{array}{ccc} -\mathrm{a}\left(\mathrm{b}^{2}+\mathrm{c}^{2}-\mathrm{a}^{2}\right) & 2 \mathrm{~b}^{3} & 2 \mathrm{c}^{3} \\ 2 \mathrm{a}^{3} & -\mathrm{b}\left(\mathrm{c}^{2}+\mathrm{a}^{2}-\mathrm{b}^{2}\right) & 2 \mathrm{c}^{3} \\ 2 \mathrm{a}^{3} & \mathrm{ab}^{3} & -\mathrm{c}\left(\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}\right) \end{array}\right|=(\mathrm{abc})\left(\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}\right)^{3}$
Solution: Taking $a, b, c$ from $C_{1}, C_{2}, C_{3}$ $=a b c\left|\begin{array}{ccc} -b^{2}-c^{2}+a^{2} & 2 b^{2} & 2 c^{2} \\ 2 a^{2} & b^{2}-c^{2}-a^{2} & 2 c^{2} \\ 2 a^{2} &...
$y=\int _{0}^{\pi /2}\frac{\sin x-\cos x}{1+\sin x\cos x}dx……(1)$
Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{0}^{\pi /2}\frac{\sin \left( \frac{\pi }{2}-x \right)-\cos \left( \frac{\pi }{2}-x \right)}{1+\sin...
$y=\int _{0}^{\pi /2}\frac{\sqrt{\frac{\sin x}{\cos x}}}{1+\sqrt{\frac{\sin x}{\cos x}}}dx$
$y=\int _{0}^{\pi /2}\frac{\sqrt{\sin x}}{\left( \sqrt{\sin x}+\sqrt{\cos x} \right)}dx......(1)$ Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ \[y=\int...
$y=\int _{0}^{\pi /2}\frac{\sqrt{\frac{\cos x}{\sin x}}}{1+\sqrt{\frac{\cos x}{\sin x}}}dx$
$y=\int _{0}^{\pi /2}\frac{\sqrt{\cos x}}{\left( \sqrt{\sin x}+\sqrt{\cos x} \right)}dx.....(1)$ Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{0}^{\pi...
$y=\int _{0}^{\pi /2}\frac{1+}{1+\sqrt{\frac{\sin x}{\cos x}}}dx$
Q15. $y=\int _{0}^{\pi /2}\frac{1+}{1+\sqrt{\frac{\sin x}{\cos x}}}dx$ $y=\int _{0}^{\pi /2}\frac{\sqrt{\cos x}}{\left( \sqrt{\sin x}+\sqrt{\cos x} \right)}dx.....(1)$ Use king theorem of definite...
$y=\int _{0}^{\pi /2}\frac{1}{1+\frac{{{\cos }^{3}}x}{{{\sin }^{3}}x}}dx$
$y=\int _{0}^{\pi /2}\frac{{{\sin }^{3}}x}{{{\sin }^{3}}x+{{\cos }^{3}}x}dx......(1)$ Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{0}^{\pi...
$y=\int _{0}^{\pi /2}\frac{1}{1+\frac{{{\sin }^{3}}x}{{{\cos }^{3}}x}}dx$
$y=\int _{0}^{\pi /2}\frac{{{\cos }^{3}}x}{{{\sin }^{3}}x+{{\cos }^{3}}x}dx......(1)$ Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{0}^{\pi...
$y=\int _{0}^{\pi /2}\frac{1}{1+\frac{\cos x}{\sin x}}dx$
$y=\int _{0}^{\pi /2}\frac{\sin x}{\sin x+\cos x}dx....(1)$ Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ \[y=\int _{0}^{\pi /2}\frac{\sin \left( \frac{\pi...
$y=\int _{0}^{\pi /2}\frac{1}{1+\frac{\sin x}{\cos x}}dx$
$y=\int _{0}^{\pi /2}\frac{\cos x}{\sin x+\cos x}dx......(1)$ Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{0}^{\pi /2}\frac{\cos \left( \frac{\pi...
Evaluate: $\int \frac{d x}{\sqrt{2 x^{2}+3 x-2}}$
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Evaluate: $\int \frac{d x}{\sqrt{x^{2}-3 x+2}}$
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Evaluate: $\int \frac{d x}{\sqrt{8+2 x-x^{2}}}$
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Evaluate: $\int\frac{d x}{\sqrt{x-x^{2}}}$
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Evaluate: $\int \frac{d x}{\sqrt{7-6 x-x^{2}}}$
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Evaluate: $\int \frac{d x}{\sqrt{16-6 x-x^{2}}}$
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Evaluate: $\int \frac{d x}{\sqrt{2+2 x-x^{2}}}$
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Evaluate: $\int \frac{d x}{\sqrt{(x-3)^{2}+1}}$
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Evaluate: $\int \frac{(x+2)}{\sqrt{x^{2}+5 x+6}} d x$
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Evaluate: $\int\frac{(6 x+5)}{\sqrt{6+x-2 x^{2}}} d x$
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Evaluate: $\int \frac{(3-2 x)}{\sqrt{2+x-x^{2}}} d x$
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Evaluate: $\int\frac{dx}{\sqrt{(2-x)^2+1}} d x$
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Evaluate: $\int \frac{(4 x+3)}{\sqrt{2 x^{2}+2 x-3}}$
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Evaluate: $\int\frac{(5 x+3)}{\sqrt{x^{2}+4 x+10}} d x$
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Evaluate: $\int \frac{(2 x+3)}{\sqrt{x^{2}+x+1}} d x$
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Evaluate: $\int \frac{x^{2}}{\sqrt{x^{6}+2 x^{3}+3}} d x$
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Evaluate: $\int \frac{d x}{\sqrt{3+4 x-2 x^{2}}}$
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Evaluate: $\int \frac{d x}{\sqrt{x} \sqrt{5-x}}$
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Evaluate: $\int\frac{dx}{\sqrt{x^{2}+6x+5}}$
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Evaluate: $\int \frac{d x}{\sqrt{1+2 x-3 x^{2}}}$
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Evaluate: $\int \frac{d x}{\sqrt{2 x^{2}+4 x+6}}$
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Evaluate: $\int {\sqrt\frac{(a-x)} {(a+x)}} d x$
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Evaluate: $\int\frac{dx}{\sqrt{1- e^{x}}}$
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Evaluate: $\int{(x-3)}{\sqrt{x^{2}+3x-18}}\;dx$
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Evaluate: $\int{(x+1)}{\sqrt{1-x-x^{2}}}\;dx$
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Evaluate: $\int{(6x+5)}{\sqrt{6+x-2x^{2}}}\;dx$
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Evaluate: $\int{(2x-5)}{\sqrt{2+3x+x^{2}}}\;dx$
As per the given question, +
Evaluate: $\int{x}\sqrt{1+x-x^{2}}\;dx$
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Evaluate: $\int(x+1) \sqrt{2 x^{2}+3}\;d x$
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Evaluate: $\int(4 x+1) \sqrt{x^{2}-x-2}\;d x$
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Evaluate: $\int(x-5) \sqrt{x^{2}+x}\;d x$
As per the given question,
Evaluate: $\int(x+2) \sqrt{x^{2}+x+1}\;d x$
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Evaluate: $\int(2 x-5) \sqrt{x^{2}-4 x+3}\;d x$
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Evaluate: $\int\sqrt{x^{2}+x+1}\;d x$
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Evaluate: $\int \sqrt{x^{2}+x}\;d x$
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$y=\int _{0}^{\pi /2}\frac{\sqrt{\frac{\cos x}{\sin x}}}{\sqrt{\frac{\sin x}{\cos x}}+\sqrt{\frac{\cos x}{\sin x}}}dx$
$y=\int _{0}^{\pi /2}\frac{\cos x}{\sin x+\cos x}dx....(1)$ Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{0}^{\pi /2}\frac{\cos \left( \frac{\pi...
$y=\int _{0}^{\pi /2}\frac{\sqrt{\frac{\sin x}{\cos x}}}{\sqrt{\frac{\sin x}{\cos xx}}+\sqrt{\frac{\cos x}{\sin x}}}dx$
$y=\int _{0}^{\pi /2}\frac{\sin x}{\sin x+\cos x}dx....(1)$ Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{0}^{\pi /2}\frac{\sin \left( \frac{\pi...
$y=\int _{0}^{\pi /2}\frac{{{\sin }^{n}}x}{{{\sin }^{n}}x+{{\cos }^{n}}x}dx….(1)$
Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{0}^{\pi /2}\frac{{{\sin }^{n}}\left( \frac{\pi }{2}-x \right)}{{{\sin }^{n}}\left( \frac{\pi }{2}-x...
$y=\int _{0}^{\pi /2}\frac{{{\sin }^{\frac{3}{2}}}x}{{{\sin }^{\frac{3}{2}}}x+{{\cos }^{\frac{3}{2}}}x}dx…..(1)$
Use king theorem of definite integral \[\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx\] \[y=\int _{0}^{\pi /2}\frac{{{\sin }^{\frac{3}{2}}}\left( \frac{\pi }{2}-x \right)}{{{\sin...
$y=\int _{0}^{\pi /2}\frac{{{\cos }^{\frac{1}{4}}}x}{{{\sin }^{\frac{1}{4}}}x+{{\cos }^{\frac{1}{4}}}x}dx…..(1)$
Use king theorem definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{0}^{\pi /2}\frac{{{\cos }^{\frac{1}{4}}}\left( \frac{\pi }{2}-x \right)}{{{\sin }^{\frac{1}{4}}}\left(...
$y=\int _{0}^{\pi /2}\frac{{{\cos }^{4}}x}{{{\sin }^{4}}x+{{\cos }^{4}}x}dx…..(1)$
Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{0}^{\pi /2}\frac{{{\cos }^{4}}\left( \frac{\pi }{2}-x \right)}{{{\sin }^{4}}\left( \frac{\pi }{2}-x...
Evaluate: $\int\sqrt{2 x^{2}+3 x+4}\;d x$
As per the given question,
$y=\int _{0}^{\pi /2}\frac{{{\cos }^{4}}x}{{{\sin }^{4}}x+{{\cos }^{4}}x}dx….(1)$
Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{0}^{\pi /2}\frac{{{\cos }^{4}}\left( \frac{\pi }{4}-x \right)}{{{\sin }^{4}}\left( \frac{\pi }{2}...
$y=\int _{0}^{\pi /2}\frac{{{\sin }^{7}}x}{{{\sin }^{7}}x+{{\cos }^{7}}x}dx…..(1)$
Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{0}^{\pi /2}\frac{{{\sin }^{7}}\left( \frac{\pi }{2}-x \right)}{{{\sin }^{7}}\left( \frac{\pi }{2}-x...
$y=\int _{0}^{\pi /2}\frac{{{\cos }^{3}}x}{{{\sin }^{3}}x+{{\cos }^{3}}x}dx……(1)$
Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{0}^{\pi /2}\frac{{{\cos }^{3}}\left( \frac{\pi }{2}-x \right)}{{{\sin }^{3}}\left( \frac{\pi }{2}-x...
Evaluate: $\int\sqrt{2\;a\;x-x^{2}}\;d x$
As per the given question,
$y=\int _{0}^{\pi /2}\frac{{{\sin }^{3}}x}{{{\sin }^{3}}x+{{\cos }^{3}}x}dx…..(1)$
Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{b}^{\pi /2}\frac{{{\sin }^{3}}\left( \frac{\pi }{2}-x \right)}{{{\sin }^{3}}\left( \frac{\pi }{2}-x...
2. $y=\int _{0}^{\pi /2}\frac{\sqrt{\sin x}}{\left( \sqrt{\sin x}+\sqrt{\cos x} \right)}dx…(1)$
Use king theorem of definite integral $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$ $y=\int _{0}^{\pi /2}\frac{\sqrt{\sin \left( \frac{\pi }{2}-x \right)}}{\left( \sqrt{\sin \left( \frac{\pi }{2}-x...
1. $y=\frac{1}{2}\int _{0}^{\frac{\pi }{2}}\frac{2\cos x}{\sin x+\cos x}dx$
$=\frac{1}{2}\int _{0}^{\frac{\pi }{2}}\frac{\cos x+\cos x-\sin x+\sin x}{\sin x+\cos x}dx$ $=\frac{1}{2}\int _{0}^{\frac{\pi }{2}}1+\frac{\cos x-\sin x}{\sin x+\cos x}dx$ $=\frac{1}{2}\left( \left(...
Evaluate: $\int\sqrt{1-4 x-x^{2}}\;d x$
As per the given question,
Evaluate: $\int\frac{2e^{x}}{\sqrt{4\;-\;e^{2} x}} d x$
As per the given question,
Evaluate: $\int\frac{e^{x}}{\sqrt{4+e^{2} x}} d x$
As per the given question,
Evaluate: $\int\sqrt{2 x-x^{2}} d x$
As per the given question,
Evaluate: $\int\sqrt{x^{2}+6 x-4} d x$
As per the given question,
Using properties of determinants prove that: $\left|\begin{array}{lll} b^{2}-a b & b-c & b c-a c \\ a b-a^{2} & a-b & b^{2}-a b \\ b c-a c & c-a & a b-a^{2} \end{array}\right|=0$
Solution: $=\left|\begin{array}{lll} b(b-a) & b-c & c(b-a) \\ a(b-a) & a-b & b(b-a) \\ c(b-a) & c-a & a(b-a) \end{array}\right|$ Taking (b-a) common from $\mathrm{C}_{1},...
Evaluate: $\int\frac{\cos x}{\sqrt{9 sin^{2} x}-1} d x$
As per the given question,
Evaluate: $\int\sqrt{x^{2}-4 x+2} d x$
As per the given question,
Using properties of determinants prove that: $\left|\begin{array}{ccc} (b+c)^{2} & a b & c a \\ a b & (a+c)^{2} & b c \\ a c & b c & (a+b)^{2} \end{array}\right|=2 a b c(a+b+c)^{3}$
Solution: $=\left|\begin{array}{ccc} b^{2}+c^{2}+2 b c & a b & a c \\ a b & a^{2}+c^{2}+2 a c & b c \\ a c & b c & a^{2}+b^{2}+2 a b \end{array}\right|$ Operating $R_{1}...
Using properties of determinants prove that: $\left|\begin{array}{lll} b^{2} c^{2} & b c & b+c \\ c^{2} a^{2} & c a & c+a \\ a^{2} b^{2} & a b & a+b \end{array}\right|=0$
Solution: Expanding with R1 $\begin{array}{l} =b^{2} c^{2}\left(a^{2} c+a b c-a b c-a^{2} b\right)-b c\left(a^{3} c^{2}+a^{2} b c^{2}-a^{2} b^{2} c-a^{3} b^{2}\right)+(b+c)\left(a^{3} b c^{2}-a^{3}...
Using properties of determinants prove that: $\left|\begin{array}{ccc} a & b-c & c+b \\ a+c & b & c-a \\ a-b & a+b & c \end{array}\right|=(a+b+c)\left(a^{2}+b^{2}+c^{2}\right)$
Solution: Operating $\mathrm{C}_{1} \rightarrow \mathrm{aCl}_{1}$ $\frac{1}{a}\left|\begin{array}{ccc} a^{2} & b-c & c+b \\ a^{2}+a c & b & c-a \\ a^{2}-a b & a+b & c...
Evaluate: $\int\cos x \sqrt{9-\sin ^{2} x} d x$
As per the given question,
Using properties of determinants prove that: $\left|\begin{array}{ccc} 1+a^{2}-b^{2} & 2 a b & -2 b \\ 2 a b & 1-a^{2}+b^{2} & 2 a \\ 2 b & -2 a & 1-a^{2}-b^{2} \end{array}\right|=\left(1+a^{2}+b^{2}\right)^{3}$
Solution: Operating $R_{1} \rightarrow R_{1}+b R_{3}, R_{2} \rightarrow R_{2^{-}} a R_{3}$ $\begin{array}{l} \left|\begin{array}{ccc} 1+a^{2}-b^{2}+2 b^{2} & 2 a b-2 a b & -2 b+b-a^{2}...
Evaluate: $\int\sqrt{3 x^{2}+4} d x$
As per the given question,
Evaluate: $\int\frac{\sin x}{\sqrt{4+\cos ^{2} x}} d x$
As per the given question,
Using properties of determinants prove that: $\left|\begin{array}{ccc} b^{2}+c^{2} & a^{2} & a^{2} \\ b^{2} & c^{2}+a^{2} & b^{2} \\ c^{2} & c^{2} & a^{2}+b^{2} \end{array}\right|=4 a^{2} b^{2} c^{2}$
Solution: $\begin{array}{l} \left|\begin{array}{ccc} b^{2}+c^{2} & a^{2} & a^{2} \\ b^{2} & c^{2}+a^{2} & b^{2} \\ c^{2} & c^{2} & a^{2}+b^{2} \end{array}\right| \\...
Evaluate: $\int \sqrt{4 x^{2}+9} d x$
As per the given question,
Using properties of determinants prove that: $\left|\begin{array}{lll} (b+c)^{2} & a^{2} & b c \\ (c+a)^{2} & b^{2} & c a \\ (a+b)^{2} & c^{2} & a b \end{array}\right|=\left(a^{2}+b^{2}+c^{2}\right)(a-b)(b-c)(c-a)(a+b+c)$
Solution: $\begin{array}{l} \left|\begin{array}{lll} (b+c)^{2} & a^{2} & b c \\ (c+a)^{2} & b^{2} & c a \\ (a+b)^{2} & c^{2} & a b \end{array}\right| \\...
Using properties of determinants prove that: $\begin{array}{l} \left|\begin{array}{ccc} (\mathrm{m}+\mathrm{n})^{2} & 1^{2} & \mathrm{mn} \\ (\mathrm{n}+1)^{2} & \mathrm{~m}^{2} & \ln \\ (1+\mathrm{m})^{2} & \mathrm{n}^{2} & \operatorname{lm} \end{array}\right|=\left(1^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}\right)(1-\mathrm{m}) \\ (\mathrm{m}-\mathrm{n})(\mathrm{n}-1) \end{array}$
Solution: $\left|\begin{array}{ccc}(\mathrm{m}+\mathrm{n})^{2} & \mathrm{l}^{2} & \mathrm{mn} \\ (\mathrm{n}+\mathrm{l})^{2} & \mathrm{~m}^{2} & \mathrm{ln} \\ (1+\mathrm{m})^{2}...
Evaluate: $\int \sqrt{x^{2}+5} d x$
As per the given question,
Evaluate: $\int\sqrt{2 x^{2}-3} d x$
As per the given question,
Evaluate: $\int\frac{\sec ^{2} x}{\sqrt{16+\tan ^{2} x}} d x$
As per the given question,
Evaluate: $\int \sqrt{x^{2}-2} d x$
As per the given question,
Evaluate: $\int\sqrt{4-9 x^{2}} d x$
As per the given question,
Evaluate: $\int \frac{3 x^{2}}{\sqrt{9-16 x^{6}}} d x$
As per the given question,
Evaluate: $\int{\sqrt{4-x^{2}}} d x$
As per the given question,
Evaluate: $\int\frac{d x}{\sqrt{9-x^{4}}}$
As per the given question,
Using properties of determinants prove that: $\left|\begin{array}{ccc} (x-2)^{2} & (x-1)^{2} & x^{2} \\ (x-1)^{2} & x^{2} & (x+1)^{2} \\ x^{2} & (x+1)^{2} & (x+2)^{2} \end{array}\right|=-8$
Solution: $\begin{array}{l} \left|\begin{array}{ccc} (\mathrm{x}-2)^{2} & (\mathrm{x}-1)^{2} & \mathrm{x}^{2} \\ (\mathrm{x}-1)^{2} & \mathrm{x}^{2} & (\mathrm{x}+1)^{2} \\...
Evaluate: $\int\frac{d x}{\sqrt{9+4x^{2}}}$
As per the given question,
Using properties of determinants prove that: $\left|\begin{array}{ccc} a^{2} & b^{2} & c^{2} \\ (a+1)^{2} & (b+1)^{2} & (c+1)^{2} \\ (a-1)^{2} & (b-1)^{2} & (c-1)^{2} \end{array}\right|=4(a-b)(b-c)(c-a)$
Solution: $\left|\begin{array}{ccc}a^{2} & b^{2} & c^{2} \\ (a+1)^{2} & (b+1)^{2} & (c+1)^{2} \\ (a-1)^{2} & (b-1)^{2} & (c-1)^{2}\end{array}\right|$...
Using properties of determinants prove that: $\left|\begin{array}{ccc} a & b & a x+b y \\ b & c & b x+c y \\ a x+b y & b x+c y & 0 \end{array}\right|=\left(b^{2}-a c\right)\left(a x^{2}+3 b x y+c y^{2}\right)$
Solution: $\left|\begin{array}{ccc} a & b & a x+b y \\ b & c & b x+c y \\ a x+b y & b x+c y & 0 \end{array}\right|$ $\begin{array}{l} \left.=\left(\frac{1}{x...
Evaluate: $\int\frac{d x}{\sqrt{1+4x^{2}}}$
As per the given question,
Using properties of determinants prove that: $\left|\begin{array}{ccc} a+b+c & -c & -b \\ -c & a+b+c & -a \\ -b & -a & a+b+c \end{array}\right|=2(a+b)(b+c)(c+a)$
Solution: $\begin{array}{l} \left|\begin{array}{ccc} a+b+c & -c & -b \\ -c & a+b+c & -a \\ -b & -a & a+b+c \end{array}\right| \\ =\left|\begin{array}{ccc} a+b & a+b &...
Using properties of determinants prove that: $\left|\begin{array}{ccc} a & a+2 b & a+2 b+3 c \\ 3 a & 4 a+6 b & 5 a+7 b+9 c \\ 6 a & 9 a+12 b & 11 a+15 b+18 c \end{array}\right|=-a^{3}$
Solution: $\left|\begin{array}{ccc}a & a+2 b & a+2 b+3 c \\ 3 a & 4 a+6 b & 5 a+7 b+9 c \\ 6 a & 9 a+12 b & 11 a+15 b+18 c\end{array}\right|$...
Using properties of determinants prove that: $\left|\begin{array}{ccc}\mathrm{b}+\mathrm{c} & \mathrm{a} & \mathrm{a} \\ \mathrm{b} & \mathrm{c}+\mathrm{a} & \mathrm{b} \\ \mathrm{c} & \mathrm{c} & \mathrm{a}+\mathrm{b} \end{array}\right|=4 \mathrm{abc}$
Solution: $\begin{array}{l} \left|\begin{array}{ccc} \mathrm{b}+\mathrm{c} & \mathrm{a} & \mathrm{a} \\ \mathrm{b} & \mathrm{c}+\mathrm{a} & \mathrm{b} \\ \mathrm{c} & \mathrm{c}...
Evaluate: $\int\frac{d x}{\sqrt{x^{2}-9}}$
As per the given question,
Using properties of determinants prove that: $\left|\begin{array}{lll} b+c & a-b & a \\ c+a & b-c & b \\ a+b & c-a & c \end{array}\right|=3 a b c-a^{3}-b^{3}-c^{3}$
Solution: $\begin{array}{l} \left|\begin{array}{lll} \mathrm{b}+\mathrm{c} & \mathrm{a}-\mathrm{b} & \mathrm{a} \\ \mathrm{c}+\mathrm{a} & \mathrm{b}-\mathrm{c} & \mathrm{b} \\...
Using properties of determinants prove that: $\left|\begin{array}{ccc} \mathrm{x} & \mathrm{y} & \mathrm{z} \\ \mathrm{x}^{2} & \mathrm{y}^{2} & \mathrm{z}^{2} \\ \mathrm{x}^{3} & \mathrm{y}^{3} & \mathrm{z}^{3} \end{array}\right|=\mathrm{xyz}(\mathrm{x}-\mathrm{y})(\mathrm{y}-\mathrm{z})(\mathrm{z}-\mathrm{x})$
Solution: $\begin{array}{l} \left|\begin{array}{ccc} \mathrm{x} & \mathrm{y} & \mathrm{z} \\ \mathrm{x}^{2} & \mathrm{y}^{2} & \mathrm{z}^{2} \\ \mathrm{x}^{3} & \mathrm{y}^{3}...
Using properties of determinants prove that: $\left|\begin{array}{ccc} 3 x & -x+y & -x+z \\ x-y & 3 y & z-y \\ x-z & y-z & 3 z \end{array}\right|=3(x+y+z)(x y+y z+z x)$
Solution: $\begin{array}{l} \left|\begin{array}{ccc} 3 x & -x+y & -x+z \\ x-y & 3 y & z-y \\ x-z & y-z & 3 z \end{array}\right| \\ =\left|\begin{array}{ccc} x+y+z & -x+y...
Using properties of determinants prove that: $\left|\begin{array}{ccc} x & x+y & x+2 y \\ x+2 y & x & x+y \\ x+y & x+2 y & x \end{array}\right|=9 y^{2}(x+y)$
Solution: $\begin{array}{l} \left|\begin{array}{ccc} \mathrm{x} & \mathrm{x}+\mathrm{y} & \mathrm{x}+2 \mathrm{y} \\ \mathrm{x}+2 \mathrm{y} & \mathrm{x} & \mathrm{x}+\mathrm{y} \\...
Using properties of determinants prove that: $\left|\begin{array}{ccc} a^{2}+2 a & 2 a+1 & 1 \\ 2 a+1 & a+2 & 1 \\ 3 & 3 & 1 \end{array}\right|=(a-1)^{3}$
Solution: $\begin{array}{l} \left|\begin{array}{ccc} a^{2}+2 a & 2 a+1 & 1 \\ 2 a+1 & a+2 & 1 \\ 3 & 3 & 1 \end{array}\right| \\ =\left|\begin{array}{ccc} a^{2}-1 & a-1...
Using properties of determinants prove that: $\left|\begin{array}{ccc} \mathrm{x}+\lambda & 2 \mathrm{x} & 2 \mathrm{x} \\ 2 \mathrm{x} & \mathrm{x}+\lambda & 2 \mathrm{x} \\ 2 \mathrm{x} & 2 \mathrm{x} & \mathrm{x}+\lambda \end{array}\right|=(5 \mathrm{x}+\lambda)(\lambda-\mathrm{x})^{2}$
Solution: $\begin{array}{l} \left|\begin{array}{ccc} \mathrm{x}+\lambda & 2 \mathrm{x} & 2 \mathrm{x} \\ 2 \mathrm{x} & \mathrm{x}+\lambda & 2 \mathrm{x} \\ 2 \mathrm{x} & 2...
Using properties of determinants prove that: $\left|\begin{array}{ccc} x+4 & 2 x & 2 x \\ 2 x & x+4 & 2 x \\ 2 x & 2 x & x+4 \end{array}\right|=(5 x+4)(x-4)^{2}$
Solution: $\left|\begin{array}{ccc}x+4 & 2 x & 2 x \\ 2 x & x+4 & 2 x \\ 2 x & 2 x & x+4\end{array}\right|$ $=\left|\begin{array}{ccc}5 \mathrm{x}+4 & 5 \mathrm{x}+4...
Using properties of determinants prove that: $\left|\begin{array}{lll} \mathrm{x} & \mathrm{a} & \mathrm{a} \\ \mathrm{a} & \mathrm{x} & \mathrm{a} \\ \mathrm{a} & \mathrm{a} & \mathrm{x} \end{array}\right|=(\mathrm{x}+2 \mathrm{a})(\mathrm{x}-\mathrm{a})^{2}$
Solution: $\begin{array}{l} \left|\begin{array}{lll} \mathrm{x} & \mathrm{a} & \mathrm{a} \\ \mathrm{a} & \mathrm{x} & \mathrm{a} \\ \mathrm{a} & \mathrm{a} & \mathrm{x}...
Evaluate: $\int\frac{d x}{\sqrt{9x^{2}-7}}$
As per the given question,
Using properties of determinants prove that: $\left|\begin{array}{ccc} \mathrm{a}+\mathrm{x} & \mathrm{y} & \mathrm{z} \\ \mathrm{x} & \mathrm{a}+\mathrm{y} & \mathrm{z} \\ \mathrm{x} & \mathrm{y} & \mathrm{a}+\mathrm{z} \end{array}\right|=\mathrm{a}^{2}(\mathrm{a}+\mathrm{x}+\mathrm{y}+\mathrm{z})$
Solution: $\begin{array}{l} \left|\begin{array}{ccc} \mathrm{a}+\mathrm{x} & \mathrm{y} & \mathrm{z} \\ \mathrm{x} & \mathrm{a}+\mathrm{y} & \mathrm{z} \\ \mathrm{x} & \mathrm{y}...
Using properties of determinants prove that: $\left|\begin{array}{ccc} 1 & 1+p & 1+p+q \\ 2 & 3+2 p & 1+3 p+2 q \\ 3 & 6+3 p & 1+6 p+3 q \end{array}\right|=1$
Solution: $\begin{array}{l} \left|\begin{array}{ccc} 1 & 1+p & 1+p+q \\ 2 & 3+2 p & 1+3 p+2 q \\ 3 & 6+3 p & 1+6 p+3 q \end{array}\right|\\ =\left|\begin{array}{ccc} -1 &...
Using properties of determinants prove that: $\left|\begin{array}{lll} 1 & \mathrm{~b}+\mathrm{c} & \mathrm{b}^{2}+\mathrm{c}^{2} \\ 1 & \mathrm{c}+\mathrm{a} & \mathrm{c}^{2}+\mathrm{a}^{2} \\ 1 & \mathrm{a}+\mathrm{b} & \mathrm{a}^{2}+\mathrm{b}^{2} \end{array}\right|=(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c})(\mathrm{c}-\mathrm{a})$
Solution: $\begin{array}{l} \left|\begin{array}{llll} 1 & \mathrm{~b}+\mathrm{c} & \mathrm{b}^{2}+\mathrm{c}^{2} \\ 1 & \mathrm{c}+\mathrm{a} & \mathrm{c}^{2}+\mathrm{a}^{2} \\ 1...
Using properties of determinants prove that: $\left|\begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ b c & c a & a b \end{array}\right|=(a-b)(b-c)(c-a)$
Solution: $\begin{array}{l} \left|\begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ b c & c a & a b \end{array}\right| \\ =\left|\begin{array}{ccc} 0 & 0 & 1 \\ a-b &...
Evaluate : $\left|\begin{array}{lll} 1^{2} & 2^{2} & 3^{2} \\ 2^{2} & 2^{2} & 4^{2} \\ 3^{2} & 4^{2} & 5^{2} \end{array}\right|$
Solution: $\left|\begin{array}{lll} 1^{2} & 2^{2} & 3^{2} \\ 2^{2} & 3^{2} & 4^{2} \\ 3^{2} & 4^{2} & 5^{2} \end{array}\right|=\left|\begin{array}{ccc} 1 & 4 & 9 \\ 4...
Evaluate: $\int\frac{d x}{\sqrt{4x^{2}-1}}$
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Evaluate : $\left|\begin{array}{ccc} 102 & 18 & 36 \\ 1 & 3 & 4 \\ 17 & 3 & 6 \end{array}\right|$
Solution: $\left|\begin{array}{ccc} 102 & 18 & 36 \\ 1 & 3 & 4 \\ 17 & 3 & 6 \end{array}\right|=6 \times\left|\begin{array}{ccc} 17 & 18 & 6 \\ 1 & 6 & 4 \\...
Evaluate: $\left|\begin{array}{lll} 29 & 26 & 22 \\ 25 & 31 & 27 \\ 63 & 54 & 46 \end{array}\right|$
Solution: $\left|\begin{array}{lll}29 & 26 & 22 \\ 25 & 31 & 27 \\ 63 & 54 & 46\end{array}\right|$ $=\left|\begin{array}{ccc}4 & -5 & -5 \\ 25 & 31 & 27 \\ 63...
Evaluate : $\left|\begin{array}{lll} 67 & 19 & 21 \\ 39 & 13 & 14 \\ 81 & 24 & 26 \end{array}\right|$
Solution: $\begin{array}{l} \left|\begin{array}{lll} 67 & 19 & 21 \\ 39 & 13 & 14 \\ 81 & 24 & 26 \end{array}\right| \\...
Evaluate: $\int\frac{d x}{\sqrt{x^{2}- 4}}$
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Evaluate: $\int\frac{d x}{\sqrt{15-8x^{2}}}$
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Evaluate: $\int\frac{d x}{\sqrt{1-9x^{2}}}$
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Evaluate: $\int\frac{d x}{\sqrt{16-x^{2}}}$
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Evaluate $\left|\begin{array}{cc}\sqrt{3} & \sqrt{5} \\ -\sqrt{5} & 3 \sqrt{3}\end{array}\right|$
Solution: $\left|\begin{array}{cc}\sqrt{3} & \sqrt{5} \\ -\sqrt{5} & 3 \sqrt{3}\end{array}\right| \cdot=3 \sqrt{3} \times \sqrt{3}-(-\sqrt{5} \times \sqrt{5})$ $=14$
Evaluate $\left|\begin{array}{cc}14 & 9 \\ -8 & -7\end{array}\right|$
Solution: $\begin{array}{l} \left|\begin{array}{cc} 14 & 9 \\ -8 & -7 \end{array}\right|=14 \times(-7)-9 \times(-8) \\ =-26 \end{array}$
For what value of $x$, the given matrix $A=\left[\begin{array}{cc}3-2 x & x+1 \\ 2 & 4\end{array}\right]$ is a singular matrix?
Solution: For $A$ to be singular matrix its determinant should be equal to 0 . $\begin{array}{l} 0=(3-2 x) \times 4-(x+1) \times 2 \\ 0=12-8 x-2 x-2 \\ 0=10-10 x \\ x=1 \end{array}$
Without expanding the determinant, prove that $\left|\begin{array}{ccc}41 & 1 & 5 \\ 79 & 7 & 9 \\ 29 & 5 & 3\end{array}\right|=0$. SINGULAR MATRIX A square matrix $A$ is said to be singular if $|A|=0$. Also, $A$ is called non singular if $|A| \neq 0$.
Solution: We know that $C_{1} \Rightarrow C_{1}-C_{2}$, would not change anything for the determinant. Applying the same in above determinant, we get $\left[\begin{array}{lll}40 & 1 & 5 \\...
Evaluate $\left|\begin{array}{lll}0 & 2 & 0 \\ 2 & 3 & 4 \\ 4 & 5 & 6\end{array}\right|$
Solution: We know that expansion of determinant with respect to first row is $a_{11} A_{11}+a_{12} A_{12}+a_{13} A_{13}$. $0(3 \times 6-5 \times 4)-2(2 \times 6-4 \times 4)+0(2 \times 5-4 \times 3)$...
Evaluate $\left|\begin{array}{ll}\cos 15^{\circ} & \sin 15^{\circ} \\ \sin 75^{\circ} & \cos 75^{\circ}\end{array}\right|$
Solution: $\begin{array}{l} \cos 15^{\circ} \cos 75^{\circ}-\sin 75^{\circ} \sin 15^{\circ} \\ =\cos \left(15^{\circ}+75^{\circ}\right) \because \cos A \cos B-\sin A \sin B=\cos (A+B) \\ =\cos...
Evaluate $\left|\begin{array}{cc}\cos 65^{\circ} & \sin 65^{\circ} \\ \sin 25^{\circ} & \cos 25^{\circ}\end{array}\right|$
Solution: By directly opening this determinant $\begin{array}{l} \cos 65^{\circ} \times \cos 25^{\circ}-\sin 25^{\circ} \times \sin 65^{\circ} \\ =\cos \left(65^{\circ}+25^{\circ}\right) \because...
Evaluate $\left|\begin{array}{ll}\sin 60^{\circ} & \cos 60^{\circ} \\ -\sin 30^{\circ} & \cos 30^{\circ}\end{array}\right|$
Solution: After finding determinant we will get, $\begin{array}{l} \operatorname{Sin} 60^{\circ}=\frac{\sqrt{3}}{2}=\cos 30^{\circ} \\ \operatorname{Cos} 60^{\circ}=\frac{1}{2}=\sin 30^{\circ} \\...
Evaluate $\left|\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right|$
Solution: After finding determinant we will get a trigonometric identity. $\begin{array}{l} \cos ^{2} \alpha+\sin ^{2} \alpha \\ =1 \end{array}$ $\because \sin ^{2} \theta+\cos ^{2} \theta=1$
Evaluate $\left|\begin{array}{cc}2 \cos \theta & -2 \sin \theta \\ \sin \theta & \cos \theta\end{array}\right|$
Solution: After finding determinant we will get a trigonometric identity. $\begin{array}{l} 2 \cos ^{2} \theta+2 \sin ^{2} \theta \\ =2 \\ \because \sin ^{2} \theta+\cos ^{2} \theta=1...
Evaluate $\left|\begin{array}{ll}\sqrt{6} & \sqrt{5} \\ \sqrt{20} & \sqrt{24}\end{array}\right|$.
Solution: Find determinant $\begin{array}{l} \sqrt{6} \times \sqrt{24-\sqrt{2}} 20 \times \sqrt{5} \\ \sqrt{1} 144-\sqrt{1} 100 \\ =12-10 \\ =2 \end{array}$
Evaluate $2\left|\begin{array}{cc}7 & -2 \\ -10 & 5\end{array}\right|$.
Solution: It is determinant multiplied by a scalar number 2 , just find determinant of matrix and multiply it by 2 . $\begin{array}{l} 2 \times(35-20) \\ 2 \times 15=30 \end{array}$
If $\mathrm{A}=\left[\begin{array}{ll}3 & 4 \\ 1 & 2\end{array}\right]$, find the value of $3|\mathrm{~A}|$.
Solution: Find the determinant of $A$ and then multiply it by 3 $\begin{array}{l} |A|=2 \\ 3|A|=3 \times 2 \\ =6 \end{array}$
If $\left|\begin{array}{cc}2 x & x+3 \\ 2(x+1) & x+1\end{array}\right|=\left|\begin{array}{ll}1 & 5 \\ 3 & 3\end{array}\right|$, write the value of $x$.
Solution: Simply by equating both sides we can get the value of $x$. $\begin{array}{l} 2 x^{2}+2 x-2\left(x^{2}+4 x+3\right)=-12 \\ \Rightarrow-6 x-6=-12 \\ \Rightarrow-6 x=-6 \\ \Rightarrow x=1...
If $\left|\begin{array}{cc}2 x & 5 \\ 8 & x\end{array}\right|=\left|\begin{array}{cc}6 & -2 \\ 7 & 3\end{array}\right|$, write the value of $x$.
Solution: This question is having the same logic as above. $\begin{array}{l} 2 x^{2}-40=18+14 \\ \Rightarrow 2 x^{2}=72 \\ \Rightarrow x^{2}=36 \\ \Rightarrow x=\pm 6 \end{array}$
If $\left|\begin{array}{cc}3 x & 7 \\ -2 & 4\end{array}\right|=\left|\begin{array}{ll}8 & 7 \\ 6 & 4\end{array}\right|$, write the value of $x$.
Solution: Here the determinant is compared so we need to take determinant both sides then find $\mathrm{x}$. $\begin{array}{l} 12 x+14=32-42 \\ \Rightarrow 12 x=-10-14 \\ \Rightarrow 12 x=-24 \\...
Evaluate $\left|\begin{array}{cc}\mathrm{a}+\mathrm{ib} & \mathrm{c}+\mathrm{id} \\ -\mathrm{c}+\mathrm{id} & \mathrm{a}-\mathrm{ib}\end{array}\right|$
Solution: This we can very simply go through directly. $\begin{array}{l} ((a+i b)(a-i b))-((-c+i d)(c+i d)) \\ \Rightarrow\left(a^{2}+b^{2}\right)-\left(-c^{2}-d^{2}\right) \\ \Rightarrow...
Evaluate $\left|\begin{array}{cc}\mathrm{x}^{2}-\mathrm{x}+1 & \mathrm{x}-1 \\ \mathrm{x}+1 & \mathrm{x}+1\end{array}\right|$
Solution: Theorem: This evaluation can be done in two different ways either by taking out the common things anc then calculating the determinants or simply take determinant. I will prefer first...
If $\mathrm{A}_{\mathrm{ij}}$ is the cofactor of the element $\mathrm{a}_{\mathrm{ij}}$ of $\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$ then write the value of $\left(\mathrm{a}_{32} \mathrm{~A}_{32}\right)$.
Solution: Theorem: $A_{i j}$ is found by deleting $j^{t h}$ rowand $j^{t h}$ column, the determinant of left matrix is called cofactor with multiplied by $(-1)^{(i+j)}$ Given: $\mathrm{j}=3$ and...
Let $A$ be a square matrix of order 3, write the value of $|2 A|$, where $|A|=4$.
Solution: Theorem: If $A$ be $k \times k$ matrix then $|p A|=p^{k}|A|$. Given: $p=2, k=3$ and $|A|=4$ $\begin{array}{l} |2 A|=2^{3} \times|A| \\ =8 \times 4 \\ =32 \end{array}$
If $A$ is a $3 \times 3$ matrix such that $|A| \neq 0$ and $|3 A|=k|A|$ then write the value of $k$.
Solution: Theorem: If Let $A$ be $k \times k$ matrix then $|p A|=p^{k}|A|$. Given: $\mathrm{k}=3$ and $\mathrm{p}=3$. $\begin{array}{l} |3 \mathrm{~A}|=3^{3} \times|\mathrm{A}| \\ =27|\mathrm{~A}|...
If $A$ is a $2 \times 2$ matrix such that $|A| \neq 0$ and $|A|=5$, write the value of $|4 A|$.
Solution: Theorem: If $A$ be $k \times k$ matrix then $|p A|=p^{k}|A|$. Given, $\mathrm{p}=4, \mathrm{k}=2$ and $|\mathrm{A}|=5$. $\begin{array}{l} |4 \mathrm{~A}|=4^{2} \times 5 \\ =16 \times 5 \\...
Evaluate: $\int\frac {dx}{(sin^{4}x+cos^{4}x)}$
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Evaluate: $\int \frac{(x^{2}+1)}{\left(x^{4}+x^{2}+1\right)} dx$
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Evaluate: $\int \frac{\left(1-x^{2}\right)}{\left(1+x^{4}\right)} d x$
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Evaluate: $\int\frac {dx} {(sin x\;-\;2\;cos x)(2\;sinx\;+\;cos x)}$
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Evaluate: $\int \frac{(2 \sin 2 \phi-\cos \phi)}{\left(6-\cos ^{2} \phi-4 \sin \phi\right)} d \phi$
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Evaluate: $\int \frac{\sin 2 x}{\left(\sin ^{4} x+\cos ^{4} x\right.} d x$
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Evaluate: $\int \frac{d x}{\left(\sin x \cos x+2 \cos ^{2} x\right)}$
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Evaluate: $\int \frac{d x}{\left(\sin ^{2} x-4 \cos ^{2} x\right)}$
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Evaluate: $\int \frac{d x}{\left(\cos ^{2} x-3 \sin ^{2} x\right)}$
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Evaluate: $\int \frac{d x}{\left(a^{2} \cos ^{2} x+b^{2} \sin ^{2} x\right)}$
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Evaluate: $\int \frac{d x}{\left (2+sin^{2}x \right)}$
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Evaluate: $\int \frac{d x}{\left (1+cos^{2}x \right)}$
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Evaluate: $\int\frac {2x}{(2+x+x^{2})} dx$
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Evaluate: $\int \frac{(\left.x-x^{2}\right)}{\left(2 x^{2}+2 x+1\right)} d x$
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Evaluate: $\int \frac{(1-3 x)}{\left(2 x^{2}+4 x+2\right)} d x$
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Evaluate: $\int \frac{x^{2}}{\left(x^{2}+6 x-3\right)} d x$
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Evaluate: $\int \frac{(2 x-3)}{\left(x^{2}+3 x-18\right)} d x$
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Evaluate: $\int \frac{(x-3)}{\left(x^{2}+2 x-4\right)} d x$
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Evaluate: $\int \frac{x}{\left(x^{2}+3 x+2\right)} d x$
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Evaluate: $\int \frac{d x}{\left(3-2 x-x^{2}\right)}$
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Evaluate: $\int \frac{d x}{\left(2 x^{2}-x-1\right)}$
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Evaluate: $\int \frac{d x}{\left(2 x^{2}+x+3\right)}$
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Evaluate: $\int \frac{d x}{\left(4 x^{2}-4 x+3\right)}$
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Evaluate: $\int \frac{d x}{\left(x^{2}+4 x+8\right)}$
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Evaluate: $\int \frac{x^{2}}{\left(a^{6}-x^{6}\right)} d x$
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Evaluate: $\int \frac{x}{\left(1-x^{4}\right)} d x$
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Evaluate: $\int \frac{d x}{\left(e^{x}+e^{-x}\right)}$
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Evaluate: $\int \frac{2 x^{3}}{\left(4+x^{8}\right)} d x$
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Evaluate:$\int \frac{3 x^{5}}{\left(1+x^{12}\right)} d x$
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Evaluate: $\int \frac{\cos x}{\left(1+\sin ^{2} x\right)} d x$
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Evaluate: $\int \frac{\sin x}{\left(1+\cos ^{2} x\right)} d x$
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Evaluate: $\int \frac{e^{x}}{\left(e^{2 x}+1\right)} d x$
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Evaluate: $\int \frac{x^{2}}{\left(9+4 x^{2}\right)} d x$
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Evaluate: $\int \frac{\left(x^{2}-1\right)}{\left(x^{2}+4\right)} d x$
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Evaluate: $\int \frac{d x}{\left(16 x^{2}-25\right)}$
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Evaluate: $\int \frac{d x}{\left(50+2 x^{2}\right)}$
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Evaluate: $\int \frac{d x}{\left(4+9 x^{2}\right)}$
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Evaluate: $\int \frac{d x}{\left(x^{2}+16\right)}$
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Evaluate: $\int \frac{d x}{\left(25-4 x^{2}\right)}$
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Evaluate: $\int \frac{d x}{(1-9 x)^{2}}$
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Evaluate: $\int \frac{1-\tan ^{2} x}{\left(1-\tan ^{2} x\right)} d x$
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Evaluate: $\int\frac {(x^{2}+1)} {(x^{4}+1)} dx$
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Evaluate: $\int\frac{2\;x\;tan^{-1} x^{2}}{\left(1+x^{4}\right)} d x$
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Evaluate: $\int(sin^{-1}x)^{2} dx$
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Evaluate: $\int\frac {dx} {x^{1/2}\;+\;x^{1/3}}$
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Evaluate: $\int\frac {sin (2 tan ^{-1}{x})} {(1+\; x^{2})} dx$
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Evaluate: $\int\frac {sin (2 tan ^{-1}{x})} {(1+\; x^{2})} dx$
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Evaluate: $\int\frac {tan x \; sec^{2}{x}}{(1-tan^{2}{x})} dx$
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Evaluate: $\int\frac {1+\; cot x}{x\;+\;log\;sinx} dx$
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Evaluate: $\int \frac {1-\;tanx} {x\;log\;cos x} dx$
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Evaluate: $\int \frac {1+sin 2x} {x+sin^{2}{x}} d x$
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Evaluate: $\int\frac{{sec}^2 (2 tan^{-1}{x})}{\left(1+x^{2}\right)} dx$
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Evaluate: $\int\frac {dx}{({x}-(\sqrt{x})}$
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Evaluate: $\int {\sqrt{{e}^{x}-1}} dx$
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Evaluate: $\int\frac {e^{\sqrt{x}} \cos \left(e^{\sqrt{x}}\right)} {{\sqrt{x}}}dx$
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