First slide

Evaluation of definite integrals

Question

The value of the integral $\int_{0}^{π / 4} \frac{\sin x + \cos x}{3 + \sin 2 x} d x$ is

Moderate

A

log 2
B

log 3
C

(1/4) log 3
D

(1/8) log 3

Solution

The integral can be written
$- \int_{0}^{π / 4} \frac{\sin x + \cos x}{(\sin x - \cos x)^{2} - 4} d x .$
Now put $t = \sin x - \cos x .$ Then $d t = (\cos x + \sin x) d x$ and the integral becomes
$\begin{array}{l} - \int_{- 1}^{0} \frac{d t}{t^{2} - 4} = - \frac{1}{4} {[\log |\frac{t - 2}{t + 2}|]}_{- 1}^{0} \\ = - \frac{1}{4} (\log 1 - \log 3) = \frac{1}{4} \log 3 . \end{array}$

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