If $f(x)$ is a function satisfying $f\left( \frac{1}{x} ight) + x^{2}f(x) = 0$ for all non-zero $x$, then $\int_{\sin heta}^{cosec\ heta}{f(x)dx}$ is equal to

Question

If $f(x)$ is a function satisfying $f\left( \frac{1}{x} ight) + x^{2}f(x) = 0$ for all non-zero $x$, then $\int_{\sin	heta}^{cosec\ 	heta}{f(x)dx}$ is equal to

Dr. Abubacker · Accepted Answer

(a)
$f\left( \frac{1}{x} ight) + x^{2}f(x) = 0$
Let $I = \int_{\sin	heta}^{cosec\ 	heta}{f(x)dx}$
Put $x = \frac{1}{t}\ \  \Rightarrow \ dx = - \frac{1}{t^{2}}\ dt$
$	herefore\ I = - \int_{{cosec}	heta}^{sin\ 	heta}{f\left( \frac{1}{t} ight)}.\frac{1}{t^{2}}\ dt$
$= \int_{\sin	heta}^{cosec\ 	heta}{\frac{1}{x^{2}}f\left( \frac{1}{x} ight)dx}$
$\Rightarrow \ I = - \int_{\sin	heta}^{cosec\ 	heta}{f(x)dx}\  \Rightarrow \ I = 0$

Q. If $f(x)$ is a function satisfying $f\left( \frac{1}{x} \right) + x^{2}f(x) = 0$ for all non-zero $x$, then $\int_{\sin\theta}^{cosec\ \theta}{f(x)dx}$ is equal to

Explanation

Reviewed by Dr. Abubacker

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