Cut Down Integral Table Memorization in Half

Last time I showed you an easy way to differentiate an inverse function. Today I will show you how to integrate an inverse function. Besides helping you remember the antiderivatives of the “arc” functions, it serves as a good integration technique in general. To date I have not seen a similar technique in any calculus book, including graduate level.

Let’s start with a function f(x). Let’s say that its antiderivative is F(x).

Now let h(x) be the inverse function of f(x). That is, f(h(x))=h(f(x))=x. If we graph f and h, we see that h is a reflection of f across the line y=x.

To integrate h(x), we use integration by parts. Recall the formula for integration by parts, plug in appropriate expressions for u and v and calculate.

However, we know from last time that

Set y=h(x), calculate, and substitute back. We arrive at the integration rule for inverse functions. Keep in mind that x=f(y), and that dx=f’(y)dy.

Let’s try an example. In this example we find the integral of arc cosine.

You can verify this with an integral table or Wolfram Integrator.

Besides using this rule to quickly remember the antiderivatives of the “arc” functions, this technique can also be used to integrate functions that cannot be integrated otherwise. Consider this example where we integrate something a lot more complicated, the arc cosine of the square root of x.