First slide

Evaluation of definite integrals

Question

$\int_{0}^{\frac{π}{4}} \frac{Sin x + Cos x}{9 + 16 Sin 2 x} d x =$

Easy

A

$\frac{1}{10} \log 3$
B

$5 \log 3$
C

$\frac{1}{20} \log 3$
D

$\log 3$

Solution

$\begin{matrix} S i n x - C o s x = t, 1 - \sin 2 x = t^{2} \\ \Rightarrow (C o s x + S i n x) d x = d t \end{matrix}$

$\int_{0}^{π / 4} \frac{Sin x + Cos x}{9 + 16 Sin 2 x} d x =$ $\frac{1}{16} \int_{- 1}^{0} \frac{1}{{(\frac{5}{4})}^{2} - t^{2}} d t$
$= \frac{1}{16} \times \frac{1}{2. \frac{5}{4}} \log {(\frac{5 + 4 t}{5 - 4 t})}_{- 1}^{0}$
$= \frac{1}{20} \log 3$

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