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

Sunday, May 13, 2012

We have to evaluate the definite integral:

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

Let $\displaystyle{t}={\arcsin{{x}}}$

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 $\displaystyle{x}=\frac{{1}}{\sqrt{{{2}}}}$ , t= $\frac{\pi}{4}$

Hence we have,

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

= $\displaystyle\frac{{\pi}^{{2}}}{{32}}$

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