Question

The value of the integral $\int \frac{\cos x}{\sin x+\cos x}dx$ is equal to

• A. $x+\log |\sin x+\cos x|+C$
• B. $\frac{1}{2}[x+\log |\sin x+\cos x|]+C$
• C. $\log |\sin x+\cos x|+C$
• D. $\frac{x}{2}+\log |\sin x+\cos x|+C$
• E. $x+\frac{1}{2}\log |\sin x+\cos x|+C$

Answer 1

The correct option is B $\frac{1}{2}[x+\log |\sin x+\cos x|]+C$

Let $l=\int \frac{\cos x}{\sin x+\cos x}dx$
$=\frac{1}{2}\int \frac{2\cos x}{\sin x+\cos x}dx$
$=\frac{1}{2}\int \frac{(\sin x+\cos x)+(\cos x-\sin x)}{(\sin x+\cos x)}dx$
$=\frac{1}{2}\int dx+\frac{1}{2}\int \frac{\cos x-\sin x}{\sin x+\cos x}\cdot dx$
$=\frac{1}{2}x+\frac{1}{2}\log |\sin x+\cos x|+C$
$=\frac{1}{2}[x+\log |\sin x+\cos x|]+C$