First slide

Evaluation of definite integrals

Question

$\int_{0}^{π} \frac{x \tan x}{S e c x + C o s x} d x =$

Moderate

A

$\frac{π}{4}$
B

$\frac{π}{2}$
C

$\frac{π^{2}}{4}$
D

$\frac{π^{2}}{2}$

Solution

$I = \int_{0}^{π} x (\frac{T a n x}{S i n x + C o s x}) d x$
    $= \frac{π + 0}{2} \int_{0}^{π} \frac{S i n x}{1 + C o s^{2} x} d x$

    $= \frac{π}{2} 2 \int_{0}^{π / 2} \frac{S i n x}{1 + C o s^{2} x} d x (∵ f (a - x) = f (x))$

    $= π {[- T a n^{- 1} (C o s x)]}_{0}^{π / 2}$

    $= - π (0 - \frac{π}{4}) = \frac{π^{2}}{4}$

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