First slide

Evaluation of definite integrals

Question

$Evaluate \int_{0}^{π / 2} x \cot x d x$

Moderate

A

0
B

2
C

3
D

4

Solution

$Integrating by parts, taking \cot x as second function, given integral becomes$
$\begin{matrix} I = [x \log \sin x]_{0}^{π / 2} - \int_{0}^{π / 2} \log \sin x d x \\ = 0 - lim_{x \to 0} (x \log \sin x) - \int_{0}^{π / 2} \log \sin x d x = \frac{1}{2} π \log 2 \end{matrix}$
$as lim_{x \to 0} x \log \sin x = lim_{x \to 0} (\frac{\log \sin x}{1 / x})$
$\begin{matrix} = lim_{x \to 0} (\frac{\cot x}{- 1 / x^{2}}) (Using L'Hopital Rule) \\ = lim_{x \to 0} (\frac{- x^{2}}{\tan x}) \\ = lim_{x \to 0} (- x \times \frac{x}{\tan x}) = 0 \times 1 = 0 \end{matrix}$

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