Question

What is the antiderivative of `arcsin(x)`?

Answer 1

`intarcsin(x)dx = xarcsin(x) + sqrt(1-x^2) + C`

Explanation:

We will be using several techniques to evaluate the given integral.

First, we use substitution :

Let `t = arcsin(x) => sin(t) = x`
Then `dx = cos(t)dt`

Making the substitution, we have

`int arcsin(x)dx = int tcos(t)dt`

`color(white)`

Next, we use integration by parts:

Let `u = t` and `dv = cos(t)dt`
Then `du = dt` and `v = sin(t)`

Applying the integration by parts formula `intudv = uv - intvdu`

`inttcos(t)dt = tsin(t) - intsin(t)dt`

`=tsin(t) - (-cos(t)) + C`

`=tsin(t)+cos(t)+C`

Finally, we substitute `x` back in. To see why `cos(t) = sqrt(1-x^2)` try drawing a right triangle in which `sin(t) = x`.

`intarcsin(x)dx = xarcsin(x)+sqrt(1-x^2)+C`