Trigonometric Integrals Broken Down to 4 Simple Cases

Trigonometric integrals can be scary. There are so many trig identities to choose from. That's why I created this page. Here we'll solve tons of examples.

To master trigonometric integrals you'll need to know the derivatives of trigonometric functions and some identities.

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We'll solve the following cases:

  • Integrals involving a sin(x) or cos(x) with at least one odd power. For example:
Here we have a trigonometric function raised to an odd power
  • Integrals involving sin(x) and cos(x) with only even powers. For example:
A trigonometric function raised to an even power
  • Integrals involving tangent and secant. For example:
An integral involving tangent
  • Just tricky trigonometric integrals that do not fall into another category. For example:
A tricky trigonometric integral. This does not fall into any other category.

Integrals With At Least One Odd Power

Let's say we want to find the integral:

An integral with an odd power

Whenever we have an odd power, we can write the integral like this:

Just write it like that to use a trigonometric identity

That is, we can write the odd power as the product of the function squared and another odd power. We do this so we would be able to use the world's most famous identity:trig

A basic trigonometric identity

Replacing this in our integrals, we get:

Substituting the identity into the integral

Doing the algebra:

Separating it into two integrals

The first integral is solved directly:

The first integral solved. How do we solve the second one?

For the remaining integral we use simple integration by substitution:

Making the substitution

And we get:

Making the substitution

This integral is easy:

Solving the integral by substitution

And finally, our integral equals:

Final answer to first example of trigonometric integral

As an exercise, you might want to solve the integral:

A similar trigonometric integral with an odd power

Answer


Integrals With Only Even Powers

Let's say we want the integral:

Trigonometric integral with even power

You might think this is an easy function to integrate. Here we need a clever substitution, though. We need to remember this identity from trigonometry:

A useful trigonometric identity

Squaring both sides, we get:

We make the even power appear in the identity

If we make a change of variables:

A change of variables

Our identity becomes:

The identity as we'll be using it

And we this identity we basically have the problem solved. Let's see how it works. Let's replace the right side of the equation in our integral:

Making use of the trig identity: we don't have exponents here (good!)

Doing the algebra:

Separating into two integrals

The first integral is solved directly. The second one we can use our trusted integration by substitution:

The first one is easy. Ideas on how to solve the second one?
Making the substitution!

Making the substitution:

Making the substitution!

And we know that integral:

This is easy!

And finally, we get:

Final answer to second example of trigonometric integrals

As an exercise, you might want to integrate:

A similar trigonometric integral with even power

Answer

To solve this one you just need to use the same identity and do the algebra.


Integrals Involving Tangent and Secant

Let's solve the integral:

A trigonometric integral involving the tangent function

To solve these integrals we'll need the identity:

A trigonometric identity that will come in handy

Here we'll use a similar method we used when we had sin(x) and cos(x) with odd powers. We can write the integral like this:

Separating things so we can use the trig identity

The only purpose of this is to use the identity I just showed you. Replacing the right side of the identity in the integral:

Replacing our trig identity inside the integral

And doing the algebra:

Separating into two integrals

Let's solve the two remaining integrals separatelly (it would be good if you are taking notes). The first one first:

Let's solve the first term

Here we can use integration by substitution, because we have the derivative of tangent sitting there.

Making a simple u substitution
Making a simple u substitution

And replacing back the u, we have our first integral:

Our first term

Now let's solve the second one:

Let's solve the second term

We can write tangent as sine over cosine:

A simple trig identity

And here again we can use simple substitution:

Using u substitution once again

Our integral becomes:

Using u substitution once again

And we get a logarithm:

That's a logarithm

Replacing back the u, we have the second integral:

Our second term

And finally, our original trigonometric integral was the difference between the first one and the second:

Final answer to third example of trigonometric integral
Final answer to third example of trigonometric integral


Just Tricky Trigonometric Integrals

There are some trigonometric integrals that simply do not fall into any of the previous cases. To solve these integrals we can only use trigonometric identities and cleverness.

As an example, let's say we want to find the integral:

We're ready for a really hairy problem

First of all we'll work on the denominator. We'll try to find a trigonometric identity to replace it. If you look closely, it looks somewhat like a perfect square.

So, we'll use the world's oldest trick: we'll add and subtract the following term:

Add and subtract this term to the denominator

Doing that, we get:

Performing an algebraic manipulation

Arranging this expression to clearly see the perfect square:

Performing an algebraic manipulation

We can factor the perfect square trinomial to get:

Factoring the perfect square we get a basic trig indentity in there

And here we use our very dear trigonometric identity to simplify this:

This became a little more simplified, but we want more

Now, we also know the following identity:

A basic trig identity

So, squaring both sides, we have:

Squaring both sides of the identity

And dividing both sides by 2:

Dividing both sides of the identity by 2

We can replace this identity in our expression for the denominator:

Replacing the previous expression in the denominator

And we can rewrite this as:

Doing just another trick

Notice that the last two expressions are equal. Just add the sines squared in the second one to get the first one.

Now, we can notice the following(this integral requires many tricks!):

Applying the basic trig identity

That is:

Applying the basic trig identity

We're arriving somewhere now. We'll multiply both sides by two:

Multiplying both sides by two

And we can "separate" the 2cos squared:

Separating the two cosines so we can use just another identity

And we use a famous identity again to get a 1:

A simpler expression

If we divide again both sides by two:

Dividing both sides by two we get a useful expression for the denominator

And this expression can be much more useful to solve the integral. Notice that until now, everything was just trigonometry. After these heroic trigonometric calculations, we're finally ready to replace it in our integral:

Replacing the new expression of the denominator into the integral

And that equals:

Doing the algebra

Now, this is much easier. We can use substitution:

Making the substitution
Making the substitution

Simplifying the mess in there:

Simplifying things

Now we can make a second substitution:

Now we make a second substitution

And our integral becomes:

This is a known integral

And in your table of integrals you may find this integral. It equals:

This is a known integral

Now we just need to replace back everything. Our t was:

Substitute back the t

And finally let's replace back the u:

Final answer to a really tricky and hairy problem

And that was hairy. I sincerely hope you don't find integrals like this one in the future. You might want to practice solving some of this kind, but don't worry if you don't find a way. They're just tricky.


Conclusion

  • Trigonometric integrals sometimes can be tricky, as there are so many trigonometric identities to choose from.
  • Most integrals involving trigonometric functions can fall into four cases: Integrals involving a sin or cos with at least one odd power, involving sin and cos with only even powers, integrals involving tangent and secant and just tricky trigonometric integrals.
  • There are specific techniques for the first three cases. For tricky integrals we just have trig identities and experience gained by practice.

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