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

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

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

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

- arcsin (x) 1 - x^{2} + x + C

Solution steps

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

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

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

Apply Integration By Parts

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

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

\int - 1 d x = - x

= - arcsin (x) 1 - x^{2} - (- x)

Simplify

= - arcsin (x) 1 - x^{2} + x

Add a constant to the solution

= - arcsin (x) 1 - x^{2} + x + C

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