Question

$\int \frac{dx}{1+2\sin x+\cos x}$ equals-

• A. $\log \left(1+2\tan x/2\right)+C$
• B. $\log \left(1-2\tan x/2\right)+C$
• C. $\frac{1}{2}\log \left(1+2\tan x/2\right)+C$
• D. None of these

Answer 1

The correct option is C $\frac{1}{2}\log \left(1+2\tan x/2\right)+C$

$I=\int \frac{1}{1+2sinx+cosx}dx$

$I=\int \frac{se{c}^2\left(\frac{x}{2}dx\right)}{2\left(2tan\right(\frac{x}{2})+1)}$

Let $t=2tan\left(\frac{x}{2}\right)+1$

$dx=\frac{1}{se{c}^2\left(\frac{x}{2}\right)}.dt$

$I=\frac{1}{2}\int \frac{1}{t}dt$

$I=\frac{1}{2}log\left(t\right)+c$

$I=\frac{1}{2}log\left(2tan\right(\frac{x}{2})+1)+c$