What is the integral of sqrt(1-x^2)arcsin(x) ?

The integral of sqrt(1-x^2)arcsin(x) is -1/4 x^2+1/4 arcsin^2(x)+1/4 arcsin(x)sin(2arcsin(x))+C

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integral of sqrt(1-x^2)arcsin(x)

\int 1 - x^{2} arcsin (x) d x

- \frac{1}{4} x^{2} + \frac{1}{4} arcsin^{2} (x) + \frac{1}{4} arcsin (x) sin (2 arcsin (x)) + C

Solution steps

\int 1 - x^{2} arcsin (x) d x

Apply Integration By Parts

: \frac{1}{2} arcsin (x) (arcsin (x) + \frac{1}{2} sin (2 arcsin (x))) - \int \frac{2 arcsin ( x ) + sin ( 2 arcsin ( x ) )}{4 1 - x ^{2}} d x

= \frac{1}{2} arcsin (x) (arcsin (x) + \frac{1}{2} sin (2 arcsin (x))) - \int \frac{2 arcsin ( x ) + sin ( 2 arcsin ( x ) )}{4 1 - x ^{2}} d x

\int \frac{2 arcsin ( x ) + sin ( 2 arcsin ( x ) )}{4 1 - x ^{2}} d x = \frac{1}{4} (x^{2} + arcsin^{2} (x))

= \frac{1}{2} arcsin (x) (arcsin (x) + \frac{1}{2} sin (2 arcsin (x))) - \frac{1}{4} (x^{2} + arcsin^{2} (x))

Simplify

\frac{1}{2} arcsin (x) (arcsin (x) + \frac{1}{2} sin (2 arcsin (x))) - \frac{1}{4} (x^{2} + arcsin^{2} (x)) : - \frac{1}{4} x^{2} + \frac{1}{4} arcsin^{2} (x) + \frac{1}{4} arcsin (x) sin (2 arcsin (x))

= - \frac{1}{4} x^{2} + \frac{1}{4} arcsin^{2} (x) + \frac{1}{4} arcsin (x) sin (2 arcsin (x))

Add a constant to the solution

= - \frac{1}{4} x^{2} + \frac{1}{4} arcsin^{2} (x) + \frac{1}{4} arcsin (x) sin (2 arcsin (x)) + C

\int \frac{e ^{x}}{1 - e ^{2 x}} d x

(3 x - 2)^{6}

\int_{0}^{1} x sin (x^{2}) d x

\int (x^{4} + 3 x)^{2} d x

x \to 1 lim (x^{3} + x^{2} - 3)

What is the integral of sqrt(1-x^2)arcsin(x) ?
The integral of sqrt(1-x^2)arcsin(x) is -1/4 x^2+1/4 arcsin^2(x)+1/4 arcsin(x)sin(2arcsin(x))+C