Question

If $\int \frac{\tan x-1}{\sec x+2\sin x}dx=\frac{\lambda }{\mu \sin x+\cos x}+C$, where $C$ is constant of integration then $(\lambda +\mu )$ is equal to

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (C)

Given integral $I=\int \frac{\frac{\sin x}{\cos x}-1}{\frac{1}{\cos x}+2\sin x}dx=\int \frac{\sin x-\cos x}{1+2\sin x\cos x}dx=\int \frac{(\sin x-\cos x)}{(\cos x+\sin x{)}^2}dx$
Let $\cos x+\sin x=t$
$=-\int \frac{1}{t^2}dt=\frac{1}{t}+C=\frac{1}{\cos x+\sin x}+C$
$\therefore \lambda +\mu =2$