Question

(1- cosx)/(1+ cosx).dx
Integrate the function

Answer 1

HELLO DEAR,


GIVEN function is integral of (1-cosx)/(1+cosx).dx


we know:- (1 - cosx) = 2 cos²(x/2),(1 + cosx) = 2sin²(x/2)

So, $\int$[2sin²(x/2)]/[2cos²(x/2].dx

=$\int$tan²(x/2).dx

=$\int${sec²(x/2)-1}.dx

=$\sf{\frac{tan(x/2)}{1/2}-x+c}$

=$\sf{2tan(x/2)-x+c}$


I HOPE ITS HELP YOU DEAR,
THANKS

Answer 2

$let \ \ I = \int {\frac{1- \cos x}{1 + \cos x}} \, dx \\ \\ I = \int {\frac{1- 1 + 2\sin^{2}\frac{x}{2}}{1 + 2\cos^{2}\frac{x}{2}-1}} \, dx \\ \\ I = \int {\frac{2\sin^{2}\frac{x}{2}}{2\cos^{2}\frac{x}{2}} \\ \\ \\ \\ I = \int {tan^{2}\frac{x}{2} \ dx \\ \\ I = \int {sec^{2}\frac{x}{2} -1 } \ dx \\ \\ I = \frac{1}2} \tan \frac{x}{2} - x +c$