What is the integral of xarcsin(pix) ?

The integral of xarcsin(pix) is 1/2 x^2arcsin(pix)-1/(4pi^2)arcsin(pix)+1/(8pi^2)sin(2arcsin(pix))+C

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integral of xarcsin(pix)

\int x arcsin (π x) d x

\frac{1}{2} x^{2} arcsin (π x) - \frac{1}{4 π ^{2}} arcsin (π x) + \frac{1}{8 π ^{2}} sin (2 arcsin (π x)) + C

Solution steps

\int x arcsin (π x) d x

Apply u-substitution

: \int \frac{u arcsin ( u )}{π ^{2}} d u

= \int \frac{u arcsin ( u )}{π ^{2}} d u

Take the constant out:

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

= \frac{1}{π ^{2}} \cdot \int u arcsin (u) d u

Apply Integration By Parts

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

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

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

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

Substitute back

u = π x

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

Simplify

\frac{1}{π ^{2}} (\frac{1}{2} (π x)^{2} arcsin (π x) - \frac{1}{4} (arcsin (π x) - \frac{1}{2} sin (2 arcsin (π x)))) : \frac{1}{2} x^{2} arcsin (π x) - \frac{1}{4 π ^{2}} arcsin (π x) + \frac{1}{8 π ^{2}} sin (2 arcsin (π x))

= \frac{1}{2} x^{2} arcsin (π x) - \frac{1}{4 π ^{2}} arcsin (π x) + \frac{1}{8 π ^{2}} sin (2 arcsin (π x))

Add a constant to the solution

= \frac{1}{2} x^{2} arcsin (π x) - \frac{1}{4 π ^{2}} arcsin (π x) + \frac{1}{8 π ^{2}} sin (2 arcsin (π x)) + C

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What is the integral of xarcsin(pix) ?
The integral of xarcsin(pix) is 1/2 x^2arcsin(pix)-1/(4pi^2)arcsin(pix)+1/(8pi^2)sin(2arcsin(pix))+C