First slide

Introduction to integration

Question

$\int \frac{1}{\sqrt{1 + \sin x}} d x =$

Moderate

A

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

$\frac{1}{\sqrt{2}} \log \tan (\frac{π}{8} + \frac{x}{4}) + c$
C

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

$\frac{1}{2 \sqrt{2}} \log \tan (\frac{π}{8} + \frac{x}{4}) + c$

Solution

$\begin{aligned} \int \frac{1}{\sqrt{1 + \sin x}} dx = \int \frac{1}{\sqrt{1 - \cos (\frac{π}{2} + x)}} dx \\ = \int \frac{1}{\sqrt{2} \sin (\frac{π}{4} + \frac{x}{2})} dx \\ = \frac{1}{\sqrt{2}} \int cosec (\frac{x}{2} + \frac{π}{4}) dx \\ = \sqrt{2} \log \tan (\frac{π}{8} + \frac{x}{4}) + c \end{aligned}$

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