1. If \[I=\int\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}dx\]
then I is equal to
a) \[\left(\sqrt{x}-2\right)\sqrt{1-x}+\cos^{-1}\sqrt{x}+C\]
b) \[\left(\sqrt{x}+2\right)\sqrt{1-x}+\sin^{-1}x+C\]
c) \[\left(2-\sqrt{x}\right)\sqrt{1-x}+\cos^{-1}x+C\]
d) \[\left(2+\sqrt{x}\right)\sqrt{1-x}-\sin^{-1}x+C\]
Explanation:
2. If \[I=\int\frac{1}{x}\sqrt{\frac{1-x}{1+x}}dx\]
then I is equal to
a) \[\log \mid x\mid+\log\mid1+\sqrt{1-x^{2}}\mid+\sin^{-1}\sqrt{x}+C\]
b) \[\log \mid x\mid-\log\mid1-\sqrt{1-x^{2}}\mid+\tan^{-1}x+C\]
c) \[\log \mid x\mid-\log\mid1+\sqrt{1-x^{2}}\mid-\sin^{-1}x+C\]
d) \[\log \mid x\mid-\log\mid1+\sqrt{1-x^{2}}\mid+\cos^{-1}x+C\]
Explanation:
3. If \[I=\int\frac{dx}{\sqrt{\left(x-\alpha\right)\left(\beta-x\right)}},\left(\beta<\alpha\right)\]
then I is equal to
a) \[\sin^{-1}\left(\frac{2x-\alpha-\beta}{\beta-\alpha}\right)+C\]
b) \[\sin^{-1}\left(\frac{x+\alpha+\beta}{\beta-\alpha}\right)\]
c) \[\sin^{-1}\left(\frac{2x+\beta-\alpha}{\alpha+\beta}\right)\]
d) \[\sin^{-1}\left(\frac{x-\alpha-\beta}{\beta-\alpha}\right)+C\]
Explanation:
4.If \[I=\int\log\left(\sqrt{1-x}+\sqrt{1+x}\right)dx\]
then I is equal to
a) \[x\log \left(\sqrt{1-x}+\sqrt{1+x}\right)+\frac{1}{2}x+C\]
b) \[x\log \left(\sqrt{1-x}+\sqrt{1+x}\right)+\frac{1}{2}\sin^{-1}+C\]
c) \[x\log \left(\sqrt{1-x}+\sqrt{1+x}\right)+\frac{1}{2}\sin^{-1}x-\frac{1}{2}x+C\]
d) \[x\log \left(\sqrt{1-x}+\sqrt{1+x}\right)+\frac{1}{2}\sin^{-1}x+\frac{1}{2}x+C\]
Explanation:
5. If \[I=\int\frac{dx}{\sin\left(x-a\right)\cos\left(x-b\right)}\]
then I is equal to
a) \[\frac{1}{\sin\left(a-b\right)}\log \mid \frac{\sin\left(x-a\right)}{\sin\left(x-b\right)}\mid +C\]
b) \[\frac{1}{\cos\left(a-b\right)}\log \mid \frac{\sin\left(x-a\right)}{\sin\left(x-b\right)}\mid +C\]
c) \[\frac{1}{\sin\left(a+b\right)}\log \mid \frac{\sin\left(x-a\right)}{\sin\left(x-b\right)}\mid +C\]
d) \[\frac{1}{\cos\left(a+b\right)}\log \mid \frac{\sin\left(x-a\right)}{\sin\left(x-b\right)}\mid +C\]
Explanation:
6. If \[I=\int\sqrt{1+\sin x}dx\]
then the domain of I is equal to
a) \[\left(-\infty ,\infty\right)\]
b) \[\left(0 ,\infty\right)\]
c) \[\left(-\infty ,0\right)\]
d) \[\left(-\pi/2,\pi/2\right)\]
Explanation:
7. If \[I=\int\frac{dx}{\left(2ax+x^{2}\right)^{3/2}}\]
then I is equal to
a) \[-\frac{x+a}{\sqrt{2ax+x^{2}}}+C\]
b) \[-\frac{1}{a}\frac{x+a}{\sqrt{2ax+x^{2}}}+C\]
c) \[-\frac{1}{a^{2}}\frac{x+a}{\sqrt{2ax+x^{2}}}+C\]
d) \[-\frac{1}{a^{3}}\frac{x+a}{\sqrt{2ax+x^{3}}}+C\]
Explanation:
8. If \[I=\int\frac{\sqrt{x^{2}+1}}{x^{4}}dx\]
then I is equal to
a) \[-\frac{1}{3}\frac{\left(x^{2}+1\right)^{3/2}}{x^{3}}+C\]
b) \[x^{3}\left(x^{2}+1\right)^{-1/2}+C\]
c) \[\frac{\sqrt{x^{2}+1}}{x^{2}}+C\]
d) \[-\frac{1}{3}\frac{\left(x^{2}+1\right)^{3/2}}{x^{2}}+C\]
Explanation:
9. If \[I=\int\frac{dx}{\sec x+ cosec x}\]
then I is equal to
a) \[\frac{1}{2}\left(\cos x+\sin x-\frac{1}{\sqrt{2}}\log\left( cosec x-\cos x\right)\right)+C\]
b) \[\frac{1}{2}\left(\sin x-\cos x-\frac{1}{\sqrt{2}}\log \mid cosec x+\cot x \mid \right)+C\]
c) \[\frac{1}{\sqrt{2}}\left(\sin x+\cos x+\frac{1}{2}\log \mid cosec x-\cos x \mid \right)+C\]
d) none of these
Explanation:
10. If \[I=\int\frac{dx}{\sqrt{1+\sin x}}\]
then I is equal to
a) \[\sqrt{2}\log\mid cosec \left(\frac{x}{2}+\frac{\pi}{4}\right)+\cot\left(\frac{x}{2}+\frac{\pi}{4}\right)\mid+C\]
b) \[\sqrt{2}\log\mid cosec \left(\frac{x}{2}+\frac{\pi}{4}\right)-\cot\left(\frac{x}{2}+\frac{\pi}{4}\right)\mid+C\]
c) \[\sqrt{2}\log\mid cosec \left(\frac{x}{2}-\frac{\pi}{4}\right)+\cot\left(\frac{x}{2}-\frac{\pi}{4}\right)\mid+C\]
d) \[\sqrt{2}\log\mid \tan\left(\frac{x}{4}-\frac{\pi}{8}\right)\mid+C\]
Explanation:
