Question

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

• A. $-1$
• B. $-2$
• C. $2$
• D. $4$

Answer 1

The correct option is C $2$

Let $I={\int }_{\pi /4}^{3\pi /4}\frac{dx}{1+cosx}...\left(i\right)$

Using, ${\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f(a+b-x)dx$

$I={\int }_{\pi /4}^{3\pi /4}\frac{dx}{1-cosx}...\left(ii\right)$

Adding Eqs.(i) and (ii)

$2I={\int }_{\pi /4}^{3\pi /4}\frac{2}{si{n}^{2}x}dx$

$\Rightarrow I={\int }_{\pi /4}^{3\pi /4}cose{c}^{2}xdx$

$I=-|cotx{|}_{\pi /4}^{3\pi /4}=2$