Theorem of Definite Integrals and its Properties
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NEET DIGITAL LIBRARY
Integral Calculus

86472 \(\int \frac{\sin x \cos x}{\sqrt{1-\sin ^{4} x}} d x=\)

1 \(\tan ^{-1}\left(\sin ^{2} \mathrm{x}\right)+\mathrm{C}\)
2 \(\tan ^{-1}(2 \sin \mathrm{x})+\mathrm{C}\)
3 \(\frac{1}{2} \sin ^{-1}\left(\sin ^{2} x\right)+C\)
4 \(\frac{1}{2} \cos ^{-1}\left(\sin ^{2} \mathrm{x}\right)+\mathrm{C}\)
Karnataka CET-2008
Integral Calculus

86473 The value of \(\int \frac{1}{1+\cos 8 x} d x\) is

1 \(\frac{\tan 2 x}{8}+C\)
2 \(\frac{\tan 8 x}{8}+C\)
3 \(\frac{\tan 4 x}{4}+C\)
4 \(\frac{\tan 4 x}{8}+C\)
Karnataka CET-2007
Integral Calculus

86475 \(\int \frac{\operatorname{cosec} x}{\cos ^{2}\left(1+\log \tan \frac{x}{2}\right)} d x=\)

1 \(\sin ^{2}\left[1+\log \tan \frac{x}{2}\right]+C\)
2 \(\tan \left[1+\log \tan \frac{x}{2}\right]+C\)
3 \(\sec ^{2}\left[1+\log \tan \frac{x}{2}\right]+C\)
4 \(-\tan \left[1+\log \tan \frac{\mathrm{x}}{2}\right]+\mathrm{C}\)
Karnataka CET-2006
Integral Calculus

86476 If \(I=\int_{\pi / 6}^{\pi / 3} \frac{d x}{1+\sqrt{\cot x}}\), then \(I=\)

1 \(\frac{\pi}{12}\)
2 \(\frac{\pi}{16}\)
3 \(\frac{\pi}{2}\)
4 \(\frac{\pi}{8}\)
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Integral Calculus

86472 \(\int \frac{\sin x \cos x}{\sqrt{1-\sin ^{4} x}} d x=\)

1 \(\tan ^{-1}\left(\sin ^{2} \mathrm{x}\right)+\mathrm{C}\)
2 \(\tan ^{-1}(2 \sin \mathrm{x})+\mathrm{C}\)
3 \(\frac{1}{2} \sin ^{-1}\left(\sin ^{2} x\right)+C\)
4 \(\frac{1}{2} \cos ^{-1}\left(\sin ^{2} \mathrm{x}\right)+\mathrm{C}\)
Karnataka CET-2008
Integral Calculus

86473 The value of \(\int \frac{1}{1+\cos 8 x} d x\) is

1 \(\frac{\tan 2 x}{8}+C\)
2 \(\frac{\tan 8 x}{8}+C\)
3 \(\frac{\tan 4 x}{4}+C\)
4 \(\frac{\tan 4 x}{8}+C\)
Karnataka CET-2007
Integral Calculus

86475 \(\int \frac{\operatorname{cosec} x}{\cos ^{2}\left(1+\log \tan \frac{x}{2}\right)} d x=\)

1 \(\sin ^{2}\left[1+\log \tan \frac{x}{2}\right]+C\)
2 \(\tan \left[1+\log \tan \frac{x}{2}\right]+C\)
3 \(\sec ^{2}\left[1+\log \tan \frac{x}{2}\right]+C\)
4 \(-\tan \left[1+\log \tan \frac{\mathrm{x}}{2}\right]+\mathrm{C}\)
Karnataka CET-2006
Integral Calculus

86476 If \(I=\int_{\pi / 6}^{\pi / 3} \frac{d x}{1+\sqrt{\cot x}}\), then \(I=\)

1 \(\frac{\pi}{12}\)
2 \(\frac{\pi}{16}\)
3 \(\frac{\pi}{2}\)
4 \(\frac{\pi}{8}\)
SRM JEE-2018
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Integral Calculus

86472 \(\int \frac{\sin x \cos x}{\sqrt{1-\sin ^{4} x}} d x=\)

1 \(\tan ^{-1}\left(\sin ^{2} \mathrm{x}\right)+\mathrm{C}\)
2 \(\tan ^{-1}(2 \sin \mathrm{x})+\mathrm{C}\)
3 \(\frac{1}{2} \sin ^{-1}\left(\sin ^{2} x\right)+C\)
4 \(\frac{1}{2} \cos ^{-1}\left(\sin ^{2} \mathrm{x}\right)+\mathrm{C}\)
Karnataka CET-2008
Integral Calculus

86473 The value of \(\int \frac{1}{1+\cos 8 x} d x\) is

1 \(\frac{\tan 2 x}{8}+C\)
2 \(\frac{\tan 8 x}{8}+C\)
3 \(\frac{\tan 4 x}{4}+C\)
4 \(\frac{\tan 4 x}{8}+C\)
Karnataka CET-2007
Integral Calculus

86475 \(\int \frac{\operatorname{cosec} x}{\cos ^{2}\left(1+\log \tan \frac{x}{2}\right)} d x=\)

1 \(\sin ^{2}\left[1+\log \tan \frac{x}{2}\right]+C\)
2 \(\tan \left[1+\log \tan \frac{x}{2}\right]+C\)
3 \(\sec ^{2}\left[1+\log \tan \frac{x}{2}\right]+C\)
4 \(-\tan \left[1+\log \tan \frac{\mathrm{x}}{2}\right]+\mathrm{C}\)
Karnataka CET-2006
Integral Calculus

86476 If \(I=\int_{\pi / 6}^{\pi / 3} \frac{d x}{1+\sqrt{\cot x}}\), then \(I=\)

1 \(\frac{\pi}{12}\)
2 \(\frac{\pi}{16}\)
3 \(\frac{\pi}{2}\)
4 \(\frac{\pi}{8}\)
SRM JEE-2018
NEET DIGITAL LIBRARY
Integral Calculus

86472 \(\int \frac{\sin x \cos x}{\sqrt{1-\sin ^{4} x}} d x=\)

1 \(\tan ^{-1}\left(\sin ^{2} \mathrm{x}\right)+\mathrm{C}\)
2 \(\tan ^{-1}(2 \sin \mathrm{x})+\mathrm{C}\)
3 \(\frac{1}{2} \sin ^{-1}\left(\sin ^{2} x\right)+C\)
4 \(\frac{1}{2} \cos ^{-1}\left(\sin ^{2} \mathrm{x}\right)+\mathrm{C}\)
Karnataka CET-2008
Integral Calculus

86473 The value of \(\int \frac{1}{1+\cos 8 x} d x\) is

1 \(\frac{\tan 2 x}{8}+C\)
2 \(\frac{\tan 8 x}{8}+C\)
3 \(\frac{\tan 4 x}{4}+C\)
4 \(\frac{\tan 4 x}{8}+C\)
Karnataka CET-2007
Integral Calculus

86475 \(\int \frac{\operatorname{cosec} x}{\cos ^{2}\left(1+\log \tan \frac{x}{2}\right)} d x=\)

1 \(\sin ^{2}\left[1+\log \tan \frac{x}{2}\right]+C\)
2 \(\tan \left[1+\log \tan \frac{x}{2}\right]+C\)
3 \(\sec ^{2}\left[1+\log \tan \frac{x}{2}\right]+C\)
4 \(-\tan \left[1+\log \tan \frac{\mathrm{x}}{2}\right]+\mathrm{C}\)
Karnataka CET-2006
Integral Calculus

86476 If \(I=\int_{\pi / 6}^{\pi / 3} \frac{d x}{1+\sqrt{\cot x}}\), then \(I=\)

1 \(\frac{\pi}{12}\)
2 \(\frac{\pi}{16}\)
3 \(\frac{\pi}{2}\)
4 \(\frac{\pi}{8}\)
SRM JEE-2018
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