find-I-e-arcsinx-dx-

Question Number 30508 by abdo imad last updated on 22/Feb/18

f i n d I = \int e^{a r c s i n x} d x .

Commented by sma3l2996 last updated on 24/Feb/18

I = \int e^{a r c s i n x} d x

u = e^{a r c s i n x} \Rightarrow u^{'} = \frac{e^{a r c s i n x}}{\sqrt{1 - x^{2}}}

v^{'} = 1 \Rightarrow v = x

I = {x e}^{a r c s i n x} - \int \frac{x}{\sqrt{1 - x^{2}}} e^{a r c s i n x} d x + c_{1}

u = e^{a r c s i n x} \Rightarrow u^{'} = \frac{e^{a r c s i n x}}{\sqrt{1 - x^{2}}}

v^{'} = \frac{x}{\sqrt{1 - x^{2}}} \Rightarrow v = - \sqrt{1 - x^{2}}

I = {x e}^{a r c s i n x} + \sqrt{1 - x^{2}} e^{a r c s i n x} - \int e^{a r c s i n x} d x + c_{2}

2 I = (x + \sqrt{1 - x^{2}}) e^{a r c s i n x} + c_{2}

I = \frac{1}{2} (x + \sqrt{1 - x^{2}}) e^{a r c s i n x} + C