What is the integral of e^{arcsin(x)} ?

The integral of e^{arcsin(x)} is 1/2 e^{arcsin(x)}x+1/2 e^{arcsin(x)}sqrt(-x^2+1)+C

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integral of e^{arcsin(x)}

\int e^{a r c s i n (x)} d x

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

Solution steps

\int e^{a r c s i n (x)} d x

Apply Integration By Parts

= e^{a r c s i n (x)} x - \int \frac{e ^{a r c s i n (x)} x}{1 - x ^{2}} d x

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

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

Simplify

e^{a r c s i n (x)} x - (- \frac{1}{2} e^{a r c s i n (x)} 1 - x^{2} + \frac{1}{2} e^{a r c s i n (x)} x) : \frac{1}{2} e^{a r c s i n (x)} x + \frac{1}{2} e^{a r c s i n (x)} - x^{2} + 1

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

Add a constant to the solution

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

What is the integral of e^{arcsin(x)} ?
The integral of e^{arcsin(x)} is 1/2 e^{arcsin(x)}x+1/2 e^{arcsin(x)}sqrt(-x^2+1)+C