Question

${\int }_{\pi /4}^{3\pi /4}\frac{dx}{1+\cos x}$ is equal to

• A. $2$
• B. $-2$
• C. $1/2$
• D. $-1/2$

Answer 1

Answer: (A)

$2$

Let $I=\int \frac{1}{1+\cos x}dx$
Substitute $t=\tan \frac{x}{2}\Rightarrow dt=\frac{1}{2}dxse{c}^{2}x$
$I=\int \frac{2}{(u^2+1)(\frac{1-u^2}{u^2+1}+1)}du=\int du=u=\tan \frac{x}{2}$
$\therefore {\int }_{\frac{\pi }{4}}^{\frac{3\pi }{4}}\frac{dx}{1+\cos x}={\left[\tan \frac{x}{2}\right]}_{\frac{\pi }{4}}^{\frac{3\pi }{4}}=2$