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Q.
If $\int \frac{\tan x-1}{\sec x+2 \sin x} d x=\frac{\lambda}{\mu \sin x+\cos x}+C$, where $C$ is constant of integration then $(\lambda+\mu)$ is equal to
Integrals
Solution:
Given integral $I=\int \frac{\frac{\sin x}{\cos x}-1}{\frac{1}{\cos x}+2 \sin x} d x=\int \frac{\sin x-\cos x}{1+2 \sin x \cos x} d x=\int \frac{(\sin x-\cos x)}{(\cos x+\sin x)^2} d x$
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 $