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.

Cut Down Derivative Table Memorization in Half

This is a useful rule that you won’t find in most books for calculating the derivative of an inverse function. This includes any of the “arc” functions in trigonometry for example.

Let’s start with a function f(x). Let’s say that its derivative 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.

Now let’s calculate h’(x).

Let’s assign the variable y=h(x). Then we can work out the derivation rule for inverse functions.

Let’s try this on a couple examples.

We know that:

Now let’s apply this rule on the inverse of f.

We also know that:

Now let’s apply this rule on the inverse of f.

We can simplify this further using the trigonometric identity

Just set

Next time I will show you the general rule for integrating an inverse function.