Question

If f(x) is a function satisfying $f\left(\frac{1}{x}\right)+x^{2}f\left(x\right)=0$ for all non-zero x, then $\underset{sin\theta }{\overset{cosec\theta }{\int }}f\left(x\right)dx$ equals to:

• A. $sin\theta +cosec\theta$
• B. $si{n}^2\theta$
• C. $cosc{e}^2\theta$
• D. None of these

Answer 1

The correct option is A $sin\theta +cosec\theta$

$letx=\left(\frac{1}{t}\right)thendx=\left(\frac{-1}{t^2}\right)dt\therefore x=sin\theta ,thent=cosec\theta andx=cosec\theta ,thent=sin\theta I={\int }_{cosec\theta }^{sin\theta }f\left(\right(\frac{1}{t}\left)\right)\left(\frac{-1}{t^2}\right)dt={\int }_{cosec\theta }^{sin\theta }-t^{2}f\left(t\right)\left(\frac{-1}{t^2}\right)dt={\int }_{cosec\theta }^{sin\theta }f\left(t\right)dt={\int }_{cosec\theta }^{sin\theta }f\left(x\right)dx\therefore 2I=0thenI=0$