Question:
Mark the correct alternative in each of the following:
$\int \frac{\cos 2 x-\cos 2 \theta}{\cos x-\cos \theta} d x$ is equal to
A. $2(\sin x+x \cos \theta)+c$
B. $2(\sin x-x \cos \theta)+C$
C. $2(\sin x+2 x \cos \theta)+C$
D. $2(\sin x-2 x \cos \theta)+C$
Solution:
$I=\int \frac{\left\{2(\cos x)^{2}-1\right\}-\left\{2(\cos \theta)^{2}-1\right\}}{\cos x-\cos \theta} d x$
$I=2 \int \frac{(\cos x)^{2}-(\cos \theta)^{2}}{\cos x-\cos \theta} d x$
$I=2 \int \frac{(\cos x-\cos \theta)(\cos x+\cos \theta)}{\cos x-\cos \theta} d x$
$I=2 \int(\cos x+\cos \theta) d x$
$I=2(\sin x+x \cos \theta)+c$