What is the integral of arcsin^2(sqrt(x)) ?

The integral of arcsin^2(sqrt(x)) is

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integral of arcsin^2(sqrt(x))

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = x

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{1}{x ^{2} + 6 x + 25} d x

\int \frac{9 x ^{2} - 6}{x ^{2}} d x

\frac{\partial}{\partial x} (\frac{- 8 x + 5}{2 y ^{2} + 1})

y^{^{'}} + \frac{1}{x} y = 7 x^{2}

\frac{d}{d x} (ln (x) - \frac{1}{x})

What is the integral of arcsin^2(sqrt(x)) ?
The integral of arcsin^2(sqrt(x)) is