First slide

Evaluation of definite integrals

Question

If $x^{2} f (x) + f (1 / x) = 0 for all x \in R ~ {0}$ then $\int_{\cos θ}^{\sec θ} f (x) d x =$

Moderate

A

$\sin 2 θ$
B

1
C

$\sec θ - \cos θ$
D

0

Solution

$I = \int_{\cos θ}^{\sec θ} f (x) d x = - \int_{\cos θ}^{\sec θ} \frac{1}{x^{2}} f (\frac{1}{x}) d x (Put \frac{1}{x} = t)$
$= \int_{\sec θ}^{\cos θ} f (t) d t = - \int_{\cos θ}^{\sec θ} f (t) d t = - I$
Hence $2 I = 0 \Rightarrow I = 0$

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