# Integral of arcsin(x)

In this post I am going to show you how to integrate the inverse function of sin(x), arcsin(x), step-by-step. But first, let’s clear a few things.

**What is arcsin(x)?**

Arcsine is the inverse function of sine, therefore:

These are equivalent forms:

The last form looks a lot like , which is equal to 1/*x*. In the case of trigonometric functions though, it just means inverse function, not one over that function. So,

but

Anyway, we are now ready to integrate!

**Integral of arcsin(x)**

The technique required for this integral is integration by parts:

Since we want to solve the integral of arcsin(x), it would make no sense to let dv=arcsin(x)dx. Also, arcsin(x) is the same as , so we can integrate 1 and differentiate arcsin(x), whose derivative is .

Now we can use u-substitution, letting 1-*x*² be our *u*. When differentiating this, it will generate an *x* that cancels out with the one at the numerator; since we already used *u* in integration by parts, we will use *t*.

Let’s now undo the substitution (remember that t=1-x²):

Overall:

And that’s it! Now try to solve this definite integral and leave the answer in the comments!

I hope everything was clear and if you have any questions leave a comment and I’ll be happy to help!