integration with a constant A

Question

Find the exact value of

$$\int^a_0 \frac{1}{x + \sqrt{a^2-x^2}}dx$$

, where a is a positive constant.

Answer 1

$$I=\int_0^a\dfrac1{x+\sqrt{a^2-x^2}}dx$$

Set $x=a\sin t$ to find $$I=\int_0^{\pi/2}\dfrac{\cos t\ dt}{\sin t+\cos t}$$

Now use the concept of Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}dx$