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. $\frac{1}{2}\left[x+log\begin{vmatrix}sin x+cos x\end{vmatrix}\right]+C$
• C. $log |sin x + cos x| + C$
• D. $\frac{x}{2}+log\begin{vmatrix}sin x+cos x\end{vmatrix}+C$
• E. $x+\frac{1}{2}+log\begin{vmatrix}sin x+cos x\end{vmatrix}+C$

Answer 1

Answer: (B)

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