1.Let \[\int\frac{x^{1/2}}{\sqrt{1-x^{3}}}dx=\frac{2}{3}gof\left(x\right)+C\]
(C being the constant of integration) then
a) \[f\left(x\right)=\sqrt{3}\]
b) \[f\left(x\right)=x^{3/2}\]
c) \[g\left(x\right)=\sin^{-1}x\]
d) Both b and c
Explanation:
2. If \[I=\int\left(\tan x\right)^{1/3}dx =A\log\frac{t^{4}-t^{2}+1}{\left(t^{2}+1\right)^{2}}+B\tan^{-1}\frac{2t^{2}-1}{2\sqrt{3}}+C\]
where \[t=\tan^{1/3}x\] then
a) A = 1/4
b) \[B=\sqrt{3}/2\]
c) A = -1/4
d) Both a and b
Explanation:
3. If \[I=\int\frac{\sin x+\sin^{3} x}{\cos 2x}dx=A\cos x+B\log \mid f\left(x\right)\mid+C\]
then
a) \[A = 1/4, B = – 1/2, f\left(x\right) =\frac{\sqrt{2}\cos x-1}{\sqrt{2}\cos x+1}\]
b) \[A = 1/2, B = -3/4 \sqrt{2}\]
c) \[A = -1/2, B = 3/\sqrt{5}, f\left(x\right) =\frac{\sqrt{2}\cos x+1}{\sqrt{2}\cos x-1}\]
d) Both b and c
Explanation:
4. The value of the integral \[\int\frac{\log\left(x+1\right)-\log x}{x\left(x+1\right)}dx\]
is
a) \[-\left(1/2\right)\left(\log x+1\right)^{2}-\left(1/2\right)\left(\log x\right)^{2}+\log\left(x+1\right) \log x+C\]
b) \[-\left[\left(\log x+1\right)^{2}-\left(\log x\right)^{2}\right]+\log\left(x+1\right) \log x+C\]
c) \[C -\left(1/2\right) (\log\left(1+1/x\right)^{2}\]
d) Both a and c
Explanation:
5. If \[\int\frac{xe^{x}}{\sqrt{1+e^{x}}}dx=f\left(x\right)\sqrt{1+e^{x}}-2\log g\left(x\right)+C\]
then
a) f (x) = x – 1
b) f (x) = 2(x – 2)
c) \[g\left(x\right)=\frac{\sqrt{1+e^{x}}-1}{\sqrt{1+e^{x}}+1}\]
d) Both b and c
Explanation:
6. If \[\int xe^{-5x^{2}}\sin4x^{2} dx=ke^{-5x^{2}}\left(A \sin 4x^{2}+B\cos4x^{2}\right)+C\]
then
a) K = – 1/82
b) K = 1/82
c) A = 5
d) Both a and c
Explanation:
7. If \[\int f\left(x\right)dx=\frac{3}{55}\sqrt[3]{\tan^{5}x}\left(5\tan^{2} x+11\right)+C\]
then f(x) is equal to
a) \[\sqrt[3]{\sin^{2} x\cos^{-14} x}\]
b) \[\sqrt[3]{\tan^{2} x\left(1+\tan^{2} x\right)^{6}}\]
c) \[\sqrt[3]{\cos^{2} x\sin^{-14} x}\]
d) Both a and b
Explanation:
8. The value of \[\int_{1}^{\sqrt{3}} \frac{1}{x^{2}}\log\sqrt{1+x^{2}}dx\]
is
a) \[\frac{\sqrt{3}-2}{2\sqrt{3}}\log 2+\frac{\pi}{8}\]
b) \[\frac{\sqrt{3}-2}{2\sqrt{3}}\log 2+\frac{\pi}{12}\]
c) \[\frac{1}{4}\log 2\]
d) \[\frac{\sqrt{3}-2}{\sqrt{3}}\log 2+\frac{\pi}{12}\]
Explanation:
9. For a, b > 0, the value of\[\int_{0}^{1} \frac{\log\frac{\left(x+a\right)^\left(x+a\right)}{\left(x+b\right)^\left(x+b\right)}}{\left(x+a\right)\left(x+b\right)\log\left(x+a\right)\log\left(x+b\right)}dx\]
is equal to
a) \[\log\left(\frac{1+a}{1+b}\right)\frac{\log a}{\log b}\]
b) \[\frac{\log\left(1+a\right)\log a}{\log\left(1+b\right)\log b}\]
c) \[\log\left(\frac{\log\left(1+a\right)\log b}{\log\left(1+b\right)\log a}\right)\]
d) \[\log\frac{\log\left(1+b\right)\log a}{\log\left(1+a\right)\log b}\]
Explanation:
10. If \[\alpha=\int_{0}^{\pi/3} \frac{dx}{1+\sin x}\]
and \[P\left(x\right)=x^{3}+2x^{2}+4\]
then the value of \[P\left(\alpha\right)\] is
a) \[\sqrt{3}+1\]
b) \[2\left(\sqrt{3}+1\right)\]
c) \[2\left(\sqrt{3}-1\right)\]
d) \[4\left(\sqrt{3}-1\right)\]
Explanation:
