1.If \[I=\int\frac{dx}{\sqrt{2x+3}+\sqrt{x+2}}\]
then I is equal to
a) \[2\left(u-v\right)+\log \mid \frac{u-1}{u+1}\mid+\log \mid \frac{v-1}{v+1}\mid+C \]
\[u=\sqrt{2x+3},v=\sqrt{x+2}\]
b) \[\log \mid \frac{\sqrt{x+2}+\sqrt{2x+3}}{\sqrt{x+2}-\sqrt{2x+3}}\mid+C\]
c) is transcedental function in u and v, \[u=\sqrt{2x+3}\] , \[v=\sqrt{x+2}\]
d) Both a and c
Explanation:
2. If \[I=\int\left[e^{x} ln \left(\sin x\right)+\frac{e^{x}}{\sin^{2}x}\right]dx\]
then I is equal to
a) \[e^{x}\left[ ln \left(\sin x\right)+\cot x\right]+C\]
b) \[e^{x}\left[ ln \left(\sin x\right)-\cot x\right]+C\]
c) \[e^{x}\left[ ln \left(\cos x\right)-\sec x\right]+C\]
d) \[e^{x}\left[ ln \left(\cos x\right)+\tan x\right]+C\]
Explanation:
3.Let \[I=\int\sqrt{\frac{cosec x-\cot x}{cosec x+\cot x}}.\frac{\sec x}{\sqrt{1+2\sec x}}dx\]
a) \[-\sqrt{2}\sin^{-1}\left(\sqrt{\frac{2}{5}}\sqrt{1-\tan^{2}\left(\frac{x}{2}\right)}\right)+C\]
b) \[\sqrt{2}\sin^{-1}\left(\sqrt{\frac{2}{5}}\sqrt{1+\tan^{2}\left(\frac{x}{2}\right)}\right)+C\]
c) \[\sqrt{2}\tan^{-1}\left(\sqrt{\frac{2}{5}}\tan\frac{x}{2}\right)+C\]
d) \[-\sqrt{2}\tan^{-1}\left(\sqrt{\frac{2}{5}}\tan\frac{x}{2}\right)+C\]
Explanation:
4. Let \[I=\int\frac{\sin^{2}x}{1+\cos^{2}x}dx\]
a) \[\frac{1}{\sqrt{2}}\tan^{-1}\left(\sqrt{2}\tan x\right)-x+C\]
b) \[\frac{1}{\sqrt{2}}\tan^{-1}\left(\tan\left( \frac{x}{2}\right)\right)-x+C\]
c) \[\frac{1}{\sqrt{2}}\tan^{-1}\left(2\tan x\right)-x+C\]
d) \[\frac{1}{\sqrt{2}}\tan^{-1}\left(\frac{1}{\sqrt{2}}\tan x\right)-x+C\]
Explanation:
5. If \[I=\int\frac{dx}{x\sqrt{x^{6}+1}}\] then
a) \[I=\frac{1}{3}ln\frac{\sqrt{x^{6}+1}-1}{\sqrt{x^{6}+1}+1}+C\]
b) \[I=\frac{1}{6}ln\frac{\sqrt{x^{6}+1}-1}{\sqrt{x^{6}+1}+1}+C\]
c) \[I=\frac{1}{3}ln\left(x^{6}+1\right)+C\]
d) \[I=\frac{1}{6}ln\left(x^{6}+1\right)+C\]
Explanation:
6. If \[I=\int\frac{10}{\left(5-x\right)^{2}}\left(\frac{5-x}{5+x}\right)^{3/4}dx\]
then I is equal to
a) \[4\left(\frac{5-x}{5+x}\right)^{1/4}+C\]
b) \[4\left(\frac{5+x}{5-x}\right)^{1/4}+C\]
c) \[2\left(\frac{5-x}{5+x}\right)^{1/4}+C\]
d) \[4\left(\frac{5-x}{5+x}\right)^{7/4}+C\]
Explanation:
7. If \[I=\int\frac{4x^{5}+5x^{4}}{\left(x^{5}+x+1\right)^{2}}dx\]
then I is equal to
a) \[\frac{x^{3}}{\left(x^{5}+x+1\right)}+C\]
b) \[\frac{x^{5}+x}{\left(x^{5}+x+1\right)}+C\]
c) \[\frac{x^{5}}{\left(x^{5}+x+1\right)}+C\]
d) \[\frac{x^{5}-x}{\left(x^{5}+x+1\right)}+C\]
Explanation:
8. Let \[I=\int\frac{x^{2}}{\left(1+x^{2}\right)\left(1+\sqrt{1+x^{2}}\right)}dx\]
is equal to
a) \[ln\left(x+\sqrt{x^{2}+1}\right)-\tan^{-1}x+C\]
b) \[\frac{1}{2}ln\left(x+\sqrt{x^{2}-1}\right)-2\tan^{-1}x+C\]
c) \[ln\left(1+\sqrt{1+x^{2}}\right)-\tan^{-1}x+C\]
d) \[ln\left(1+\sqrt{1+x^{2}}\right)+\tan^{-1}x+C\]
Explanation:
9. For a > 1, let \[I=\int a^{x}\left[ln x +\left(ln a \right)ln \left(\frac{x^{x}}{a^{x}}\right)\right]dx\]
then I is equal to
a) \[a^{x}\left[x ln \left(\frac{x}{a}\right) +1-\frac{1}{ln a}\right]+C\]
b) \[a^{x}\left[x ln \left(\frac{x}{a}\right) -\frac{1}{ln a}\right]+C\]
c) \[a^{x}\left[x ln \left(\frac{x}{a}\right) +1\right]+C\]
d) \[a^{x}\left[ ln \left(\frac{x}{a}\right) +x-\frac{1}{ln a}\right]+C\]
Explanation:
10. Let \[I=\int\left(\sqrt{\frac{x}{4-x}}-\sqrt{\frac{4-x}{x}}\right)dx\]
then
a) \[\left(4-x\right)^{3/2}-x^{3/2}+\sqrt{4-x}-\sqrt{x}+C\]
b) \[x^{3/2}-\left(4-x\right)^{3/2}+\sqrt{x}+C\]
c) \[-\sqrt{4x-x^{2}}+C\]
d) \[-\sin^{-1}\left(\frac{x-2}{2}\right)+C\]
Explanation:
