`int xarcsinx dx` Find the indefinite integral

Recall that indefinite integral follows `int f(x) dx = F(x) +C` where:

`f(x)` as the integrand function

`F(x)` as the antiderivative of `f(x)`

`C` as the constant of integration.

The given integral problem: `int x arcsin(x)dx `  resembles a formula from integration table. We follow the integral formula for inverse sine...

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Recall that indefinite integral follows `int f(x) dx = F(x) +C` where:

`f(x)` as the integrand function

`F(x)` as the antiderivative of `f(x)`

`C` as the constant of integration.

The given integral problem: `int x arcsin(x)dx `  resembles a formula from integration table. We follow the integral formula for inverse sine function as:

`int x arcsin(ax) dx = (x^2arcsin(ax))/2-arcsin(ax)/(4a^2)+(xsqrt(1-a^2x^2))/(4a)+C`

Applying the integral formula inverse sine function with `a=1` , we get:

`int x arcsin(x) dx = (x^2arcsin(1*x))/2-arcsin(1*x)/(4*1^2)+(xsqrt(1-1^2x^2))/(4*1)+C`

`= (x^2arcsin(x))/2-arcsin(x)/4+(xsqrt(1-x^2))/4+C`

or  `((2x^2-1)arcsin(x)+xsqrt(1-x^2))/4+C`

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