First slide

Evaluation of definite integrals

Question

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

Difficult

A

$\sin θ + cosec θ$
B

$\sin^{2} θ$
C

${cosec}^{2} θ$
D

None of these

Solution

we have, $f (\frac{1}{x}) + x^{2} f (x) = 0 \Rightarrow f (x) = - \frac{1}{x^{2}} f (\frac{1}{x})$
$\begin{matrix} I = \int_{\sin θ}^{cosec θ} f (x) dx = \int_{\sin θ}^{cosec θ} \{- \frac{1}{x^{2}} f (\frac{1}{x})\} dx \\ = \int_{cosec θ}^{\sin θ} f (t) dt, where t = \frac{1}{x} \\ \Rightarrow I = - \int_{\sin θ}^{cosec θ} f (t) dt = - I \Rightarrow 2 I = 0 \Rightarrow I = 0 \end{matrix}$

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