First slide

Evaluation of definite integrals

Question

$\int_{0}^{\frac{π}{4}} (π x - 4 x^{2}) \ln (1 + \tan x) d x =$

Moderate

A

$\frac{π^{2}}{96} \ln 2$
B

$\frac{π^{3}}{192} \ln 2$
C

$\frac{π^{2}}{96}$
D

$\frac{π^{3}}{192}$

Solution

$I = 4 \int_{0}^{\frac{π}{4}} x (\frac{π}{4} - x) \ln (1 + \tan x) d x, x \to \frac{π}{4} - x$
$I = 4 \int_{0}^{\frac{π}{4}} (\frac{π}{4} - x) x \ln (1 + \tan (\frac{π}{4} - x)) d x$

$I = 4 \int_{0}^{\frac{π}{4}} x (\frac{π}{4} - x) x \ln (\frac{2}{1 + \tan x}) d x$

$2 I = 4 \ln 2 \int_{0}^{\frac{π}{4}} x (\frac{π}{4} - x) d x$

$I = 2 \ln 2 [{(\frac{π}{4})}^{3} \cdot \frac{1}{2} - {(\frac{π}{4})}^{3} \cdot \frac{1}{3}]$ $= \frac{π^{3}}{192} \ln 2$

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