What is the integral of xarcsin(x^2) ?

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

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integral of xarcsin(x^2)

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

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

Solution steps

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

Apply Integration By Parts

: \frac{1}{2} x^{2} arcsin (x^{2}) - \int \frac{x ^{3}}{1 - x ^{4}} d x

= \frac{1}{2} x^{2} arcsin (x^{2}) - \int \frac{x ^{3}}{1 - x ^{4}} d x

\int \frac{x ^{3}}{1 - x ^{4}} d x = - \frac{1}{2} 1 - x^{4}

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

Simplify

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

Add a constant to the solution

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

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