What is the integral of xrcsin(x) ?

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

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integral of xrcsin(x)

\int x d arcsin (x) d x

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

Solution steps

\int x d arcsin (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 arcsin (x) d x

Apply Integration By Parts

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

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

\int e^{x} sech (x) d x

\int \frac{3 x + 3}{x ^{2} + 8 x + 16} d x

\int sec^{2} (π x) + e^{- 3 x} d x

\int sec^{2} (2 a) x d x

\frac{1}{u}

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