Trigonometric Substitution Intuition You'll Hardly Find on Your Textbook

As you may know by now, indefinite integration is quite easy. Trigonometric substitution is one of the tricks you need to learn to solve integrals.

This is in fact a really clever trick. You may start watching this video:


Let's start with an example. Let's say we want to find the integral:

First example of trigonometric substitution

In fact, this integral is quite easy. There is more than one way to solve it. We'll try our trigonometric substitution with it.

Here we can't use simple integration by substitution. We don't have an x sitting in the numerator. The denominator looks like a trigonometric identity, though. Just imagine the x were a sin(x) or cos(x). It smells trigonometric, right?

Let's remember this from trigonometry:

Fundamental trigonometric identity

If we subtract sin squared of θ from both sides:

Fundamental trigonometric identity

Now, our denominator looks a lot like the right side of this identity. So, we'll try and use the substitution:

Making the trigonometric substitution
Making the trigonometric substitution

We have our x, equal to sin(θ). Now, to make this substitution we need to get dx. Deriving x with respect to θ:

Finding the other terms of the integral

So, our dx is:

Differential in trigonometric substitution

Replacing this in our integral:

Making the trigonometric substitution

Now, the purpose of doing that substitution was to use the trigonometric identity I showed you at the beginning. Let's do that. We have the identity:

Making use of trigonometric identities in integral
Making use of trigonometric identities in integral

We can replace this in our denominator:

Simplifying things in integral by trigonometric substitution

And our integral simplifies nicely:

Simplifying things in integral by trigonometric substitution

And are we done? Not yet! Our original integral was in terms of x. So, our answer should also be a function of x. We made the substitution:

Substituting back in trigonometric substitution

This means that:

Substituting back in trigonometric substitution

This is just the definition of the inverse trigonometric functions. So, finally, our integral is:

Final answer to example 1 of trigonometric substitution


Why Trigonometric Substitution Works

In the previous example we solved an integral that had a radical. Our goal with trigonometric substitution is always to simplify the radicals.

This is possible because of the trigonometric identities:

Useful trigonometric identities for integrals

Generally, we can have three cases which we can solve using trigonometric substitution.


First Case

We have in the integral the expression:

First case of trigonometric substitution

    Here a squared is a constant and x is the variable. In this case we can use the first identity involving sines and cosines, because we have a difference.

    How do we do that? We factor the constant:

Solving first case of trigonometric substitution

And we make the change of variable:

Making the trigonometric substitution for first case

So, our radical becomes:

Making the trigonometric substitution for first case

And this usually makes the integral easier to solve.


Second Case

The second case is when we have an integral with the expression:

Second case of trigonometric substitution

Here we'll use the second identity involving tangent and secant, because we have a sum.

Again, we follow the same steps. First, we factor the constant:

Solving second case of trigonometric substitution

And we make the substitution:

Making the trigonometric substitution for second case

So, we get:

Making the trigonometric substitution for second case


Third Case

Third case of trigonometric substitution

Here we use a variation of the second identity:

Trigonometric identity for third case of trigonometric substitution

Here again we factor the constant:

Solving third case of trigonometric substitution

And we use the substitution:

Making the substitution for third case of trigonometric substitution

And we get:

Making the substitution for third case of trigonometric substitution

And that's all trigonometric substitution is about. Let's do another example...


Example 2

Watch this example solved on the video:



Let's find the integral:

Example 2 of trigonometric substitution

As you may have seen, this integral falls into the third case. We will follow all the process. Let's factor the constant 16:

Solving example 2 of trigonometric substitution

And now, let's use the substitution:

Making the substitution for example 2 of trigonometric substitution

So, our x is:

Making the substitution for example 2 of trigonometric substitution

Our x squared:

Making the substitution for example 2 of trigonometric substitution

We also need to substitute dx. To get that, we derive x with respect to θ:

Making the substitution for example 2 of trigonometric substitution

And you may find at a table of derivatives that:

Making the substitution for example 2 of trigonometric substitution

This is easily proved using what we learned in derivatives of trigonometric functions.

So, the derivative of x is:

Making the substitution for example 2 of trigonometric substitution

Substituting all this in our integral:

Making the substitution for example 2 of trigonometric substitution

Here we use the trigonometric identity to substitute the denominator:

Cleaning up the mess in example 2 of trigonometric substitution

We simplify a bit and finally we have the integral:

Apparently simpler integral in example 2 of trigonometric substitution

Ok, I haven't proved this to you yet, but this integral is:

Formula for integral of secant cubed

This is proved using integration by parts. This integral is quite tricky and I will pay special attention to it in a future page.

Let's accept this fact now, to get:

Applying the formula of integral of secant cubed in example 2

Now, we need to substitute back the secants and tangents. Let's remember what our substitution was:

Substituting back the variables in example 2 of trigonometric substitution

So, putting secant in function of x:

Substituting back the variables in example 2 of trigonometric substitution

So, we have:

Substituting back the variables in example 2 of trigonometric substitution

Now, substituting back:

Substituting back the variables in example 2 of trigonometric substitution
Substituting back the variables in example 2 of trigonometric substitution

And finally, our integral is:

Final answer to example 2 of trigonometric substitution

And that was a hairy problem. This page is long enough. In the links below you'll find more examples of trigonometric substitution.

Conclusion

  • Trigonometric substitution is not hard. It is just a trick used to find primitives.
  • It is usually used when we have radicals within the integral sign.
  • There are three basic cases, and each follow the same process. The only difference between them is the trigonometric substitution we use.
  • Remember to find dx as a function of θ before you actually make the substitution!



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