First slide

Evaluation of definite integrals

Question

For a continuous function $f$ , the value $\int_{0}^{\infty} f (x^{n} + x^{- n}) \frac{\log x}{x} + \frac{1}{1 + x^{2}} d x$ is

Moderate

A

$\frac{π}{2}$
B

0
C

$- π$
D

$2 π$

Solution

Putting $\frac{1}{x} = t \Rightarrow d x = - \frac{1}{t^{2}} d t$ , so
$\begin{array}{l} I = \int_{0}^{\infty} f (x^{n} + x^{- n}) \log x \frac{d x}{x} \\ = - \int_{\infty}^{0} f (t^{- n} + t^{n}) (\log \frac{1}{t}) t \frac{d t}{t^{2}} \\ = - \int_{0}^{\infty} f (t^{- n} + t^{n}) \log t \frac{d t}{t} = - I \\ \Rightarrow 2 I = 0 \Rightarrow I = 0 \end{array}$
The given integral is equal to
$\int_{0}^{\infty} \frac{1}{1 + x^{2}} d x = {\tan^{- 1} x|}_{0}^{\infty} = \frac{π}{2}$

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