How to Integrate trigonometric functions. Dear friends, today I will show how to integrate trigonometric functions. Have a look!!

In general, we all have studied integration during high school. So this is more like a re-visit to the good old topic.

### How to Integrate trigonometric functions

In one of my earlier posts, I have already talked about how to integrate any function with the help of substitution.

Today I’ll show you how to integrate trigonometric functions.

For that, I’ll use some standard formulas in trigonometry a lot.

As I said before, it’s an old topic from high school. We all know well about it.

But still, I bring up this topic because we have to use integration a lot in engineering mathematics.

For example, to solve any first order differential equation, we must know integration. Otherwise, it won’t work.

And there are many more examples like that.

So now I’ll give you some examples.

#### Some examples of how to integrate trigonometric functions

Disclaimer: None of these examples is mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here are the examples.

##### Example 1

According to Stroud and Booth (2013)*

##### Solution

Here the integrand is .

That means I have to integrate this function.

Let

Now from the standard differentiation formulas of trigonometric functions, I already know that

I’ll use this formula in this example.

How?

Have a look!

Here goes the first step.

###### Step 1

I already know that

Here I’ll try to write in terms of .

So that I can use the differentiation formula

Now I will rewrite the integrand as

Next I’ll write in terms of .

We all know that the function is nothing but

Therefore the integrand will be

As a next step, I’ll substitute with any arbitrary value, say, .

Then I’ll do the integration.

###### Step 2

So if I choose as , then I can replace with the function of .

Let me choose

Now I’ll differentiate both sides with respect to .

Then it will be

This means

Thus I can say

Now I can replace with . Also I’ll substitute with .

So this will be

Next, I’ll do the integration.

###### Step 3

Here I’ll use one of the standard formulas of integration. That is,

Therefore in this problem, it will be

Here is the integration constant.

Now I’ll replace with .

So it will look like

This is the answer to this example.

Now comes the second example.

##### Example 2

According to Stroud and Booth (2013)*

##### Solution

Here the integrand is .

That means I have to integrate this function.

Let

Here again, I will use two of the standard formulas in trigonometry.

These are

and

If I add these two formulas, it gives

That means

Now in this example, I will use this formula to simplify the integrand.

So that will be my first step.

###### Step 1

Here I’ll rewrite the integrand as a sum of two sine functions.

I already know from the formula

Hence I’ll replce with and with .

Thus will be

Now I’ll simplify it a bit to get

The next step is quite straight forward.

I’ll use the standard integration formulas to get the value of .

Therefore it will be

Hence I can conclude that this is the answer to the given example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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