Integration by Parts
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Integral Calculus

86328 If \(\int \frac{\cos x d x}{\sin ^{3} x\left(1+\sin ^{6} x\right)^{2 / 3}}=f(x)\left(1+\sin ^{6} x\right)^{1 / \lambda}+C\)
where \(C\) is a constant of integration, then is equal \(t\)

1 2
2 \(-\frac{9}{8}\)
3 -2
4 \(\frac{9}{8}\)
JEE Main-2020-08.01.2020
Integral Calculus

86329 The integral \(\int \frac{d x}{(x+4)^{8 / 7}(x-3)^{6 / 7}}\) is equal to (where \(C\) is a constant of integration)

1 \(-\left(\frac{x-3}{x+4}\right)^{-1 / 7}+C\)
2 \(\frac{1}{2}\left(\frac{x-3}{x+4}\right)^{3 / 7}+C\)
3 \(\left(\frac{x-3}{x+4}\right)^{1 / 7}+C\)
4 \(\frac{1}{13}\left(\frac{x-3}{x+4}\right)^{-13 / 7}+C\)
JEE Main-2020-09.01.2020
Integral Calculus

86330 The integral
\(\int \frac{e^{3 \log _{e} 2 x}+5 e^{2 \log _{e} 2 x}}{e^{4 \log _{e} x}+5 e^{3 \log _{e} x}-7 e^{2 \log _{e} x}} d x \quad x>0\), is equal to (where, \(\boldsymbol{c}\) is a constant of integration)

1 \(\log _{e}\left|\mathrm{x}^{2}+5 \mathrm{x}-7\right|+\mathrm{c}\)
2 \(4 \log _{e}\left|x^{2}+5 x-7\right|+c\)
3 \(\frac{1}{4} \log _{\mathrm{e}}\left|\mathrm{x}^{2}+5 \mathrm{x}-7\right|+\mathrm{c}\)
4 \(\log _{e} \sqrt{x^{2}+5 x-7}+c\)
JEE Main-2021-25.02.2021
Integral Calculus

86331 Integrate \(\frac{\sec ^2\left(\sin ^{-1} x\right)}{\sqrt{1-x^2}}\)

1 \(\sin \left(\tan ^{-1} x\right)+c\)
2 \(\tan \left(\sec ^{-1} x\right)+c\)
3 \(\tan \left(\sin ^{-1} x\right)+c\)
4 \(-\tan \left(\cos ^{-1} x\right)+c\)
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Integral Calculus

86328 If \(\int \frac{\cos x d x}{\sin ^{3} x\left(1+\sin ^{6} x\right)^{2 / 3}}=f(x)\left(1+\sin ^{6} x\right)^{1 / \lambda}+C\)
where \(C\) is a constant of integration, then is equal \(t\)

1 2
2 \(-\frac{9}{8}\)
3 -2
4 \(\frac{9}{8}\)
JEE Main-2020-08.01.2020
Integral Calculus

86329 The integral \(\int \frac{d x}{(x+4)^{8 / 7}(x-3)^{6 / 7}}\) is equal to (where \(C\) is a constant of integration)

1 \(-\left(\frac{x-3}{x+4}\right)^{-1 / 7}+C\)
2 \(\frac{1}{2}\left(\frac{x-3}{x+4}\right)^{3 / 7}+C\)
3 \(\left(\frac{x-3}{x+4}\right)^{1 / 7}+C\)
4 \(\frac{1}{13}\left(\frac{x-3}{x+4}\right)^{-13 / 7}+C\)
JEE Main-2020-09.01.2020
Integral Calculus

86330 The integral
\(\int \frac{e^{3 \log _{e} 2 x}+5 e^{2 \log _{e} 2 x}}{e^{4 \log _{e} x}+5 e^{3 \log _{e} x}-7 e^{2 \log _{e} x}} d x \quad x>0\), is equal to (where, \(\boldsymbol{c}\) is a constant of integration)

1 \(\log _{e}\left|\mathrm{x}^{2}+5 \mathrm{x}-7\right|+\mathrm{c}\)
2 \(4 \log _{e}\left|x^{2}+5 x-7\right|+c\)
3 \(\frac{1}{4} \log _{\mathrm{e}}\left|\mathrm{x}^{2}+5 \mathrm{x}-7\right|+\mathrm{c}\)
4 \(\log _{e} \sqrt{x^{2}+5 x-7}+c\)
JEE Main-2021-25.02.2021
Integral Calculus

86331 Integrate \(\frac{\sec ^2\left(\sin ^{-1} x\right)}{\sqrt{1-x^2}}\)

1 \(\sin \left(\tan ^{-1} x\right)+c\)
2 \(\tan \left(\sec ^{-1} x\right)+c\)
3 \(\tan \left(\sin ^{-1} x\right)+c\)
4 \(-\tan \left(\cos ^{-1} x\right)+c\)
J&K CET-2014
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Integral Calculus

86328 If \(\int \frac{\cos x d x}{\sin ^{3} x\left(1+\sin ^{6} x\right)^{2 / 3}}=f(x)\left(1+\sin ^{6} x\right)^{1 / \lambda}+C\)
where \(C\) is a constant of integration, then is equal \(t\)

1 2
2 \(-\frac{9}{8}\)
3 -2
4 \(\frac{9}{8}\)
JEE Main-2020-08.01.2020
Integral Calculus

86329 The integral \(\int \frac{d x}{(x+4)^{8 / 7}(x-3)^{6 / 7}}\) is equal to (where \(C\) is a constant of integration)

1 \(-\left(\frac{x-3}{x+4}\right)^{-1 / 7}+C\)
2 \(\frac{1}{2}\left(\frac{x-3}{x+4}\right)^{3 / 7}+C\)
3 \(\left(\frac{x-3}{x+4}\right)^{1 / 7}+C\)
4 \(\frac{1}{13}\left(\frac{x-3}{x+4}\right)^{-13 / 7}+C\)
JEE Main-2020-09.01.2020
Integral Calculus

86330 The integral
\(\int \frac{e^{3 \log _{e} 2 x}+5 e^{2 \log _{e} 2 x}}{e^{4 \log _{e} x}+5 e^{3 \log _{e} x}-7 e^{2 \log _{e} x}} d x \quad x>0\), is equal to (where, \(\boldsymbol{c}\) is a constant of integration)

1 \(\log _{e}\left|\mathrm{x}^{2}+5 \mathrm{x}-7\right|+\mathrm{c}\)
2 \(4 \log _{e}\left|x^{2}+5 x-7\right|+c\)
3 \(\frac{1}{4} \log _{\mathrm{e}}\left|\mathrm{x}^{2}+5 \mathrm{x}-7\right|+\mathrm{c}\)
4 \(\log _{e} \sqrt{x^{2}+5 x-7}+c\)
JEE Main-2021-25.02.2021
Integral Calculus

86331 Integrate \(\frac{\sec ^2\left(\sin ^{-1} x\right)}{\sqrt{1-x^2}}\)

1 \(\sin \left(\tan ^{-1} x\right)+c\)
2 \(\tan \left(\sec ^{-1} x\right)+c\)
3 \(\tan \left(\sin ^{-1} x\right)+c\)
4 \(-\tan \left(\cos ^{-1} x\right)+c\)
J&K CET-2014
NEET DIGITAL LIBRARY
Integral Calculus

86328 If \(\int \frac{\cos x d x}{\sin ^{3} x\left(1+\sin ^{6} x\right)^{2 / 3}}=f(x)\left(1+\sin ^{6} x\right)^{1 / \lambda}+C\)
where \(C\) is a constant of integration, then is equal \(t\)

1 2
2 \(-\frac{9}{8}\)
3 -2
4 \(\frac{9}{8}\)
JEE Main-2020-08.01.2020
Integral Calculus

86329 The integral \(\int \frac{d x}{(x+4)^{8 / 7}(x-3)^{6 / 7}}\) is equal to (where \(C\) is a constant of integration)

1 \(-\left(\frac{x-3}{x+4}\right)^{-1 / 7}+C\)
2 \(\frac{1}{2}\left(\frac{x-3}{x+4}\right)^{3 / 7}+C\)
3 \(\left(\frac{x-3}{x+4}\right)^{1 / 7}+C\)
4 \(\frac{1}{13}\left(\frac{x-3}{x+4}\right)^{-13 / 7}+C\)
JEE Main-2020-09.01.2020
Integral Calculus

86330 The integral
\(\int \frac{e^{3 \log _{e} 2 x}+5 e^{2 \log _{e} 2 x}}{e^{4 \log _{e} x}+5 e^{3 \log _{e} x}-7 e^{2 \log _{e} x}} d x \quad x>0\), is equal to (where, \(\boldsymbol{c}\) is a constant of integration)

1 \(\log _{e}\left|\mathrm{x}^{2}+5 \mathrm{x}-7\right|+\mathrm{c}\)
2 \(4 \log _{e}\left|x^{2}+5 x-7\right|+c\)
3 \(\frac{1}{4} \log _{\mathrm{e}}\left|\mathrm{x}^{2}+5 \mathrm{x}-7\right|+\mathrm{c}\)
4 \(\log _{e} \sqrt{x^{2}+5 x-7}+c\)
JEE Main-2021-25.02.2021
Integral Calculus

86331 Integrate \(\frac{\sec ^2\left(\sin ^{-1} x\right)}{\sqrt{1-x^2}}\)

1 \(\sin \left(\tan ^{-1} x\right)+c\)
2 \(\tan \left(\sec ^{-1} x\right)+c\)
3 \(\tan \left(\sin ^{-1} x\right)+c\)
4 \(-\tan \left(\cos ^{-1} x\right)+c\)
J&K CET-2014
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