Integration by substitution. Dear friends, today’s topic is integration by substitution. In general, we all have studied integration during high school.

So this is more like a re-visit to the good old topic. Have a look!!

### Integration by substitution

Today I’ll talk about one of the most used methods of integration. And, that is, ‘integration by substitution’.

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.

#### Examples of the integration by substitution

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

Here I will use the substitution method to integrate this function.

But there are also several other ways to do it as well. But I find this one as the simplest one.

So here goes the first step.

###### Step 1

Now from the standard formulas in differentiation, I already know that .

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 2

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 I’ll go to the next example.

##### Example 2

According to Stroud and Booth (2013)*

##### Solution

In this example the integrand is .

Let

Here again I will use the substitution method to integrate this function.

###### Step 1

First of all, I will divide both top and bottom of with .

Then it will look like

Now I’ll simplify it.

From the standard trigonometry formulas, we all know that

Thus the integrand will be

As a next step, I’ll replace with .

###### Step 2

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

Therefore in this example, first I’ll write in the form of

So it will be

Now as per the standard formula, the value of will be

Here is the integration constant.

Now I’ll replace with .

So it will look like

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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