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

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

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integral of xarcsin(2x^2)

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

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

Solution steps

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

Apply u-substitution

= \int \frac{arcsin ( u )}{4} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{4} \cdot \int arcsin (u) d u

Apply Integration By Parts

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

\int \frac{u}{1 - u ^{2}} d u = - 1 - u^{2}

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

Substitute back

u = 2 x^{2}

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

Simplify

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

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

Add a constant to the solution

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

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