Question

How to integrate arc sin x dx?

Answer 1

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

Explanation:

We will proceed by using integration by substitution and integration by parts.

Substitution:

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

Then, substituting, we have

`intarcsin(x)dx = inttcos(t)dt`

Integration by Parts:

Let `u = t` and `dv = cos(t)dt`

Then `du = dt` and `v = sin(t)`

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

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

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

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

`=arcsin(x)*sin(arcsin(x))+cos(arcsin(x))+C`

As `sin(arcsin(x)) = x` and `cos(arcsin(x)) = sqrt(1-x^2)`

(try drawing a right triangle where `sin(theta)=x` and calculate `cos(theta)` to obtain the second equality)

we obtain our final result:

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