# Integral of arccos(x)

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

**What is arccos(x)?**

Arccosine is the inverse function of cosine, therefore:

These are equivalent forms:

The last form looks a lot like x^-1, 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 arccos(x)**

The technique required for this integral is integration by parts:

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

Your final line had a typo. You forgot to multiply arccos(x) by x which was the v value

Fixed! Thank you!