What is the integral of arcsin(2x+1) ?

The integral of arcsin(2x+1) is 1/2 (arcsin(2x+1)(2x+1)+2sqrt(-x(x+1)))+C

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integral of arcsin(2x+1)

\int arcsin (2 x + 1) d x

\frac{1}{2} (arcsin (2 x + 1) (2 x + 1) + 2 - x (x + 1)) + C

Solution steps

\int arcsin (2 x + 1) d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = 2 x + 1

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

Simplify

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

= \frac{1}{2} (arcsin (2 x + 1) (2 x + 1) + 2 - x (x + 1))

Add a constant to the solution

= \frac{1}{2} (arcsin (2 x + 1) (2 x + 1) + 2 - x (x + 1)) + C

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What is the integral of arcsin(2x+1) ?
The integral of arcsin(2x+1) is 1/2 (arcsin(2x+1)(2x+1)+2sqrt(-x(x+1)))+C