Question

Solve $\frac{1}{\sqrt{3}}\int \frac{dx}{\sin (x+\frac{\pi }{4})+1}$.

Answer 1

Let $I=\frac{1}{\sqrt{3}}\int \frac{dx}{\sin (x+\frac{\pi }{4})+1}$

Let $(x+\frac{\pi }{4})=t\Rightarrow dx=dt$

$\therefore I=\frac{1}{\sqrt{3}}\int \frac{dt}{1+\sin t}$

$=\frac{1}{\sqrt{3}}\int \frac{1-\sin t}{{\cos }^{2}t}dt=\frac{1}{\sqrt{3}}\int ({\sec }^{2}t-\sec t\tan t)dt$

$=\frac{1}{\sqrt{3}}[\tan t-\sec t]+c$

$=\frac{1}{\sqrt{3}}[\tan (x+\frac{x}{4})-\sec (x+\frac{x}{4})]+c$