KVPY Sample Paper KVPY Stream-SX Model Paper-31

Let\[g(x)=\] in \[x\] and \[f(x)=\left( \frac{1-x\cos x}{1+x\cos x} \right)\] then \[\int\limits_{\frac{-\pi }{4}}^{\frac{\pi }{4}}{g(f(x))}dx\] is equal to:

A) \[\ell n1\]

B) \[\ell n2\]

C) \[\ell ne\]

D) \[\ell n4\]

Correct Answer: A

Solution :

\[g(x)=In(x)\]

\[f(x)=\frac{1-x\cos x}{1+x\cos x}\] and \[g(f(x)=\ell n\left( \frac{1-x\cos x}{1+x\cos x} \right)\]

\[I=\int\limits_{-\pi /4}^{\pi /4}{\ell n\left( \frac{1+x\cos x}{1-x\cos x} \right)}dx\]

Adding, \[2I=\int\limits_{-\pi /4}^{\pi /4}{\ell n(1)}dx=0\]

\[\Rightarrow \] \[I=0.\]