`int_0^(1/sqrt(2)) arcsinx/sqrt(1-x^2) dx` Evaluate the definite integral

We have to evaluate the definite integral:

`\int_{0}^{1/\sqrt{2}}\frac{arc sinx}{\sqrt{1-x^2}}dx`

Let `t= arcsinx`

Differentiating both sides we get,

`\frac{1}{\sqrt{1-x^2}}dx=dt`                 (Since we know that `\frac{d}{dx}(arc sinx)=\frac{1}{\sqrt{1-x^2}}`

Now when x=0, t=0

and when      `x=1/sqrt(2)` , t= `\frac{\pi}{4}`   

Hence we have,

`\int_{0}^{1/\sqrt{2}}\frac{arc sinx}{\sqrt{1-x^2}}dx=\int_{0}^{\pi/4}tdt`

                      =`pi^2/32`