First slide

Introduction to integration

Question

The value of $\int \frac{\cos x}{\sin x + \cos x} d x$ is

Easy

A

$\frac{x^{2}}{2} + \log | \sin x + \cos x | + C$
B

$\frac{1}{2} (x + \log | \sin x + \cos x |) + C$
C

$2 \log | \sin x + \cos x | + x + C$
D

none of these

Solution

$\int \frac{\cos x}{\sin x + \cos x} d x$
$\begin{aligned} = \frac{1}{2} \int \frac{\cos x + \sin x + \cos x - \sin x}{\sin x + \cos x} d x \\ = \frac{1}{2} \int (1 + \frac{\cos x - \sin x}{\sin x + \cos x}) d x \\ = (1 / 2) (x + \log | \sin x + \cos x |) + C \end{aligned}$

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