1.For \[x\epsilon R\] Let \[f\left(x\right)=\cos\left(\sin x\right)\cos^{2}\frac{x}{2}+\sin\left(\sin x\right)\sin^{2}\frac{x}{2}\]
if \[2\int e^{x}f\left(x\right)dx=e^{x}\left[h\left(u\left(x\right)\right)+u\left(u\left(x\right)\right)\right]+C\]
then
a) h(x) = cos x
b) u'(x) = h(x)
c) \[u (2\pi) = 0\]
d) All of the Above
Explanation:
2. If \[I=\int\frac{dx}{\left(x+5\right)^{5}+\left(x+7\right)^{3}}=a\left(f\left(x\right)\right)^{b}+C\]
then
a) 2ab = – 1
b) 32a + b = 0
c) \[f\left(x\right)=\frac{x+7}{x+5}\]
d) All of the Above
Explanation:
3.If \[I_{n}=\int\cot^{n}x dx\] and \[I_{0}+I_{1}+2\left(I_{2}+....+I_{8}\right)+I_{9}+I_{10}=A\left(u+\frac{u^{2}}{2}+....+\frac{u^{9}}{9}\right)+\]
constant where
u = cot x then
a) A is constant
b) A = – 1
c) A = 1
d) Both a and b
Explanation:
4. If\[\int\frac{\sin x}{\sin\left( x-\pi/4\right)}dx= A\left(f\left(x\right)+\log\mid sin x-\cos x\mid\right)+C\]
then
a) \[A=\sqrt{2}\]
b) \[A=1/\sqrt{2}\]
c) f (x) = x
d) Both b and c
Explanation:
5. If \[f'\left(x\right)=\frac{1}{-x+\sqrt{x^{2}+1}}\] and \[f\left(0\right)=-\frac{1+\sqrt{2}}{2}\]
then
f(1) is equal to
a) \[-\log\left(\sqrt{2}-1\right)\]
b) 1
c) log (1+ \[\sqrt{2}\] )
d) Both a and c
Explanation:
6. If \[I=\int\log\left(\sqrt{x-a}+\sqrt{x-b}\right)dx\]
then I is equal to
a) \[[2x-\left(a+b\right)\log\left(\sqrt{x-a}+\sqrt{x-b}\right)+C\]
b) \[\frac{1}{2}\left[2x-\left(a+b\right)\right]\left(\log\left(b-a\right)\right)ln\left(\sqrt{x-a}-\sqrt{x-b}\right)-\left(\frac{1}{2}\right)\sqrt{\left(x-a\right)\left(x-b\right)}+C\]
c) \[\frac{1}{2}\left[2x-\left(a+b\right)\right]\log\left(\sqrt{x-a}+\sqrt{x-b}\right)-\frac{1}{2}\sqrt{\left(x-a\right)\left(x-b\right)}+C\]
d) Both b and c
Explanation:
7. If \[I=\int\frac{x^{2}+20}{\left(x\sin x+5\cos x\right)^{2}}dx\]
then I is equal to
a) \[-\frac{x}{\cos x\left(x\sin x+5\cos x\right)}+\tan x+C\]
b) \[\frac{x}{\cos x\left(x\sin x+5\cos x\right)}+\cot x+C\]
c) \[\left(x\sin x-5\cos x\right)^{-1}\sin x+7x+C\]
d) \[\frac{x}{\cos x\left(x\sin x+5\cos x\right)}+2\tan x+C\]
Explanation:
8. If \[I=\int\frac{5x^{8}+7 x^{6}}{\left(x^{2}+1+2x^{7}\right)^{2}}dx\]
then I is equal to
a) \[\frac{x^{7}}{2x^{7}+x^{2}+1}+C\]
b) \[\frac{x^{5}}{x^{2}+1+2x^{7}}+C\]
c) \[\frac{p\left(x\right)}{q\left(x\right)}\] , deg p(x) = deg q(x) = 7
d) Both a and c
Explanation:
9. If \[I=\int\frac{\sin^{3}\left(\theta/2\right)}{\cos\left(\theta/2\right)\sqrt{\cos^{3}\theta+\cos^{2}\theta+\cos\theta}}d\theta\]
then I is equal to
a) \[\cot^{-1}\left(\tan\theta+\sec\theta\right)+Const\]
b) \[\cot^{-1}\left(\cos\theta+\sec\theta+1\right)+Const\]
c) \[\tan^{-1}\left(\cos\theta+\sec\theta+1\right)+Const\]
d) Both b and c
Explanation:
10. If \[I=\int e^{x\sin x+\cos x}\left(\frac{x^{4}\cos^{3}x-x\sin x+\cos x}{x^{2}\cos^{2}x}\right)dx\]
then I is equal to
a) \[ e^{x\sin x+\cos x}\left(x-\frac{\sec x}{x}\right)+C\]
b) \[ e^{x\sin x+\cos x}\left(x\sin x-\frac{\cos x}{x}\right)\]
c) \[ xe^{x\sin x+\cos x}-\int e^{x\sin x+\cos x} \left(1-\frac{\cos x-x\sin x}{x^{2}\cos^{2}x}\right)dx\]
d) Both a and c
Explanation:
