Question

If $f\left(x\right)$ is a function satisfying, $f\left(\frac{1}{x}\right)+x^{2}f\left(x\right)=0$ for all non-zero $x$,

then evaluate ${\int }_{\sin \theta }^{\csc \theta }f\left(x\right)dx$.

Answer 1

We have, $f\left(\frac{1}{x}\right)+x^{2}f\left(x\right)=0$
$\Rightarrow f\left(x\right)=-\frac{1}{x^2}f\left(\frac{1}{x}\right)$
Let $I={\int }_{\sin \theta }^{\csc \theta }f\left(x\right)dx$
$={\int }_{\sin \theta }^{\csc \theta }-\frac{1}{x^2}f\left(\frac{1}{x}\right)dx$
Put $\frac{1}{x}=t$
$\Rightarrow -\frac{1}{x^2}dx=dt$
$={\int }_{\csc \theta }^{\sin \theta }f\left(t\right)dt$
$=-{\int }_{\sin \theta }^{\csc \theta }f\left(t\right)dt=-I$ (Integration is independent of change of variable)
$\therefore 2I=0⟹I=0$