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.
We'll solve the following cases:
Let's say we want to find the integral:
Whenever we have an odd power, we can write the integral like this:
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
Replacing this in our integrals, we get:
Doing the algebra:
The first integral is solved directly:
For the remaining integral we use simple integration by substitution:
And we get:
This integral is easy:
And finally, our integral equals:
As an exercise, you might want to solve the integral:
Let's say we want the integral:
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:
Squaring both sides, we get:
If we make a change of variables:
Our identity becomes:
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:
Doing the algebra:
The first integral is solved directly. The second one we can use our trusted integration by substitution:
Making the substitution:
And we know that integral:
And finally, we get:
As an exercise, you might want to integrate:
To solve this one you just need to use the same identity and do the algebra.
Let's solve the integral:
To solve these integrals we'll need the identity:
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:
The only purpose of this is to use the identity I just showed you. Replacing the right side of the identity in the integral:
And doing the algebra:
Let's solve the two remaining integrals separatelly (it would be good if you are taking notes). The first one first:
Here we can use integration by substitution, because we have the derivative of tangent sitting there.
And replacing back the u, we have our first integral:
Now let's solve the second one:
We can write tangent as sine over cosine:
And here again we can use simple substitution:
Our integral becomes:
And we get a logarithm:
Replacing back the u, we have the second integral:
And finally, our original trigonometric integral was the difference between the first one and the second:
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:
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:
Doing that, we get:
Arranging this expression to clearly see the perfect square:
We can factor the perfect square trinomial to get:
And here we use our very dear trigonometric identity to simplify this:
Now, we also know the following identity:
So, squaring both sides, we have:
And dividing both sides by 2:
We can replace this identity in our expression for the denominator:
And we can rewrite this as:
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!):
We're arriving somewhere now. We'll multiply both sides by two:
And we can "separate" the 2cos squared:
And we use a famous identity again to get a 1:
If we divide again both sides by two:
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:
And that equals:
Now, this is much easier. We can use substitution:
Simplifying the mess in there:
Now we can make a second substitution:
And our integral becomes:
And in your table of integrals you may find this integral. It equals:
Now we just need to replace back everything. Our t was:
And finally let's replace back the u:
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.
If you have just a general doubt about a concept, I'll try to help you. If you have a problem, or set of problems you can't solve, please send me your attempt of a solution along with your question. These will appear on a new page on the site, along with my answer, so everyone can benefit from it.
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