Question

Evaluate $\int \frac{dx}{si{n}^{2}x+sin2x}$.

• A. $\frac{1}{2}ln\left|\frac{tanx}{tanx+2}\right|+C$
• B. $\frac{1}{2}ln\left|\frac{cotx}{cotx+2}\right|+C$
• C. $\frac{1}{2}ln\left|\frac{tanx}{tanx-2}\right|+C$
• D. $\frac{1}{2}ln\left|\frac{cotx+2}{cotx}\right|+C$

Answer 1

The correct option is A $\frac{1}{2}ln\left|\frac{tanx}{tanx+2}\right|+C$

$I=\int \frac{dx}{si{n}^{2}x+sin2x}$

Multiply numerator and denominator by ${\sec }^{2}x$

$I=\int \frac{se{c}^{2}xdx}{ta{n}^{2}x+2tanx}$

[Substitute $tanx=t$]$⟹se{c}^{2}xdx=dt$

$I=\int \frac{dt}{t^2+2t}$

$=\frac{1}{2}\int \left(\frac{1}{t}-\frac{1}{t+2}\right)dt$

$=\frac{1}{2}\left[ln\right(t)-\ln (t+2\left)\right]+C$

$=\frac{1}{2}\left[ln\right(\tan x)-\ln (\tan x+2\left)\right]+C$

$=\frac{1}{2}ln\left|\frac{tanx}{tanx+2}\right|+C$