Question

Value of $I={\int }_0^a\frac{dx}{x+\sqrt{a^2-x^2}}$ is

• A. $\pi /4$
• B. $\pi /2$
• C. $\pi$
• D. $2\pi$

Answer 1

The correct option is A $\pi /4$

$I={\int }_0^a\frac{dx}{x+\sqrt{a^2-x^2}}$

Substituting $x=asint\Rightarrow dx=costdt$

$I={\int }_0^{\frac{\pi }{2}}\frac{costdt}{sint+cost}$ ...(1)

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

$I={\int }_0^{\frac{\pi }{2}}\frac{sintdt}{cost+sint}$ ...(2)

Adding (1) and (2), we get

$2I={\int }_0^{\frac{\pi }{2}}dt=\frac{\pi }{2}\Rightarrow I=\frac{\pi }{4}$