First slide

Introduction to integration

Question

$\int \frac{d x}{\sin x + \cos x}$ is equal to

Easy

A

$\log \tan (π / 4 + x / 8) + C$
B

$\sqrt{2} \log \tan (x / 2 + π / 8) + C$
C

$\frac{1}{\sqrt{2}} \log |\tan (\frac{x}{2} + \frac{π}{8})| + C$
D

$\sqrt{2} \log \tan (\frac{x}{4} - \frac{π}{8}) + C$

Solution

$\begin{array}{l} \int \frac{d x}{\sin x + \cos x} \\ = \frac{1}{\sqrt{2}} \int \frac{d x}{\sin (π / 4) \sin x + \cos (π / 4) \cos x} \\ = \frac{1}{\sqrt{2}} \int \frac{d x}{\cos (x - π / 4)} = \frac{1}{\sqrt{2}} \int \sec (x - π / 4) d x \\ = \frac{1}{\sqrt{2}} \log |\tan (\frac{π}{4} + \frac{x}{2} - \frac{π}{8})| + C \\ = \frac{1}{\sqrt{2}} \log |\tan (\frac{π}{8} + \frac{x}{2})| + C \end{array}$

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