What is the integral of xrccos(x) ?

The integral of xrccos(x) is d(1/2 x^2arccos(x)+1/4 (arcsin(x)-1/2 sin(2arcsin(x))))+C

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integral of xrccos(x)

\int x d arccos (x) d x

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

Solution steps

\int x d arccos (x) d x

Take the constant out:

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

= d \cdot \int x arccos (x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of xrccos(x) ?
The integral of xrccos(x) is d(1/2 x^2arccos(x)+1/4 (arcsin(x)-1/2 sin(2arcsin(x))))+C