1.If \[I=\int\frac{dx}{x^{1/2}+x^{1/3}}\]
then I is equal to
a) \[2\sqrt{x}-3x^{1/3}+6x^{1/6}-\log\left(1+x^{1/6}\right)+C\]
b) \[3x^{1/3}-6x^{1/6}-\log\left(1+x^{1/6}\right)+C\]
c) \[\sqrt{x}+3x^{1/3}-6x^{1/6}+\log\left(1+x^{1/6}\right)+C\]
d) \[3x^{1/3}-\sqrt{x}+6x^{1/6}-\log\left(1+x^{1/6}\right)+C\]
Explanation:
2. If \[I=\int\frac{\left(3\sin\theta-2\right)\cos\theta}{5-\cos^{2}\theta-4\sin\theta}d\theta\]
then I is equal to
a) \[3\log \left(2-\sin\theta\right)+\frac{4}{2-\sin\theta}+C\]
b) \[3\log \mid2-\sin\theta\mid+\frac{4}{2+\sin\theta}+C\]
c) \[3\log \left(2+\sin\theta\right)-\frac{2}{2+\cos\theta}+C\]
d) \[3\log (2+\sin\theta)-\frac{2}{2-\sin\theta}+C\]
Explanation:
3.If \[I=\int\frac{dx}{4+5\sin x}\]
then I is equal to
a) \[\frac{1}{3}\log\mid\frac{\tan\left(x/2\right)+5}{3} \mid+C\]
b) \[\frac{1}{3}\log\mid\frac{2\tan\left(x/2\right)+1}{\tan\left(x/2\right)+2} \mid+C\]
c) \[\frac{1}{3}\log\mid\frac{2\tan\left(x/2\right)+1}{\tan\left(x/2\right)-2} \mid+C\]
d) \[\frac{1}{\sqrt{3}}\log\mid\frac{2\tan\left(x/2\right)-1}{\tan\left(x/2\right)+1} \mid+C\]
Explanation:
4. If \[I=\int\frac{x^{3}+x}{x^{4}-9}dx\]
then I is equal to
a) \[\frac{1}{4}\log \mid x^{4}-9 \mid+\frac{1}{12}\log \mid \frac{x^{2}+3}{x^{2}-3}\mid +C\]
b) \[\frac{1}{4}\log \mid x^{4}-9 \mid-\frac{1}{12}\log \mid \frac{x^{2}-3}{x^{2}+3}\mid +C\]
c) \[\frac{1}{4}\log \mid x^{4}-9 \mid-\frac{1}{12}\log \mid \frac{x-3}{x+3}\mid +C\]
d) \[\frac{1}{2}\log \mid x^{4}-9 \mid-\frac{1}{6}\log \mid \frac{x^{2}-3}{x^{2}+3}\mid +C\]
Explanation:
5. If \[I=\int e^{x} \left(x \cos x+\sin x\right) dx\]
then I is equal to
a) \[\frac{1}{2} e^{x} \left(x \sin x-\cos x\right) +C\]
b) \[\frac{1}{2} e^{x} \left(x \sin x+\cos x\right) +C\]
c) \[\frac{1}{2} e^{x} \left(x \cos x-\sin x\right) +C\]
d) \[\frac{1}{2} e^{x}\left(x \left(\sin x+\cos x\right)-\cos x\right) +C\]
Explanation:
6. If \[I=\int\sqrt{e^{2x}+ae^{x}}dx\]
then I is equal to
a) \[\sqrt{e^{2x}+ae^{x}}+a\log \left(e^{x}+ae^{2x}\right)+C\]
b) \[\sqrt{e^{2x}+ae^{x}}+a\log \left(e^{x/2}+a^{ex}\right)+C\]
c) \[\sqrt{e^{2x}+ae^{x}}+a\log \left(e^{x/2}+\sqrt{e^{x}+a}\right)+C\]
d) \[\sqrt{e^{2x}+ae^{x}}+\log \left(e^{x/2}+\sqrt{e^{x}+e^{x/2}}\right)+C\]
Explanation:
7. If \[I=\int\frac{\tan x}{\sqrt{a+b\tan^{2} x}}dx \left(a< b\right)\]
then I is equal to
a) \[\frac{1}{\sqrt{b-a}}\sin^{-1}\left(\sqrt{\frac{a+b\tan^{2} x}{b-a}}\right)+C\]
b) \[\frac{1}{\sqrt{b-a}}\cos^{-1}\left(\sqrt{\frac{a+b\tan^{2} x}{b-a}}\right)+C\]
c) \[\frac{1}{\sqrt{b-a}}\tan^{-1}\left(\sqrt{\frac{a+b\tan^{2} x}{b-a}}\right)+C\]
d) \[\frac{1}{\sqrt{b-a}}\tan^{-1}\left(\sqrt{\frac{b-a}{a+b\tan^{2} x}}\right)+C\]
Explanation:
8. If \[I=\int\frac{dx}{\sin x\left(3+2\cos x\right)}\]
then I is equal to
a) \[\left(2/5\right)\log\left(3+2\cos x\right)-\left(1/2\right)\log\left(1-\cos x\right)+\left(1/10\right)\log\left(1+\cos x\right)+C\]
b) \[\left(2/5\right)\log\left(3+2\cos x\right)+\left(1/2\right)\log\left(1-\cos a\right)-\left(1/10\right)\log\left(1+\cos x\right)+C\]
c) \[\left(2/5\right)\log\left(3+2\cos x\right)-\left(1/2\right)\log\left(1+\cos x\right)+\left(1/10\right)\log\left(1-\cos x\right)+C\]
d) \[\left(2/5\right)\log\left(3+2\cos x\right)+\left(1/10\right)\log\sin x+C\]
Explanation:
9. If \[I=\int\frac{\sin 2x}{\left(3+4\cos x\right)^{3}}dx\]
then I is equal to
a) \[\frac{3\cos x+8}{\left(3+4\cos x\right)^{2}}+C\]
b) \[\frac{3+8\cos x}{16\left(3+4\cos x\right)^{2}}+C\]
c) \[\frac{3+\cos x}{\left(3+4\cos x\right)^{2}}+C\]
d) \[\frac{3-8\cos x}{16\left(3+4\cos x\right)^{2}}+C\]
Explanation:
10. If \[I=\int e^{-x}\log\left(e^{x}+1\right)dx\]
then I is equal to
a) \[x+\left(e^{-x}+1\right)\log\left(e^{x}+1\right)+C\]
b) \[x+\left(e^{x}+1\right)\log\left(e^{x}+1\right)+C\]
c) \[x-\left(e^{-x}+1\right)\log\left(e^{x}+1\right)+C\]
d) none of these
Explanation:
