What is the integral of xarccos(2x) ?

The integral of xarccos(2x) is 1/2 x^2arccos(2x)+1/16 (arcsin(2x)-1/2 sin(2arcsin(2x)))+C

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integral of xarccos(2x)

\int x arccos (2 x) d x

\frac{1}{2} x^{2} arccos (2 x) + \frac{1}{16} (arcsin (2 x) - \frac{1}{2} sin (2 arcsin (2 x))) + C

Solution steps

\int x arccos (2 x) d x

Apply Integration By Parts

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

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

\int - \frac{x ^{2}}{1 - 4 x ^{2}} d x = - \frac{1}{16} (arcsin (2 x) - \frac{1}{2} sin (2 arcsin (2 x)))

= \frac{1}{2} x^{2} arccos (2 x) - (- \frac{1}{16} (arcsin (2 x) - \frac{1}{2} sin (2 arcsin (2 x))))

Simplify

= \frac{1}{2} x^{2} arccos (2 x) + \frac{1}{16} (arcsin (2 x) - \frac{1}{2} sin (2 arcsin (2 x)))

Add a constant to the solution

= \frac{1}{2} x^{2} arccos (2 x) + \frac{1}{16} (arcsin (2 x) - \frac{1}{2} sin (2 arcsin (2 x))) + C

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What is the integral of xarccos(2x) ?
The integral of xarccos(2x) is 1/2 x^2arccos(2x)+1/16 (arcsin(2x)-1/2 sin(2arcsin(2x)))+C