Integration by parts. Hello friends, today I’ll show how it works. Let’s start now!!

Want to know more about the integration? Do check out:

**How to integrate trigonometric functions**

**Integration by partial fractions**

**Integration by parts**

Suppose I have a function like and I have to integrate it. Then I’ll use the technique of integration by parts. According to this technique, one function will be and the other function will be .

So the technique is:

Now I’ll give you some examples to explain myself better.

**Solved examples of integration by parts**

Disclaimer: None of these examples is mine. I have chosen these from some book or books. The references are at the end of the post.

**Example 1**

According to Croft et al. (2000) “Use integration by parts to find the following:

”

**Solution**

Let

Now I can also rewrite it as

As I can see, Here I have to integrate a product of two functions and . So I can use the method ‘integration by parts’. Now here I will choose as since is . And then I’ll be able to integrate this expression successfully. So it will be

and this gives

Next, I’ll simplify it to get

Then I’ll integrate it further to get

Again I’ll simplify it to get

So this gives

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

Now I’ll give another example.

**Example 2**

According to Croft et al. (2000) “Use integration by parts to find:

”

**Solution**

Let

As I can see, Here I have to integrate a product of two functions and . So I can use the method ‘integration by parts’. Now here I will choose as since is . And then I’ll be able to integrate this expression successfully. So it will be

and this gives

Next, I’ll simplify it. And that gives

Then I’ll integrate it further to get

Here is the integration constant.

Now I’ll simplify . So this means

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

Now I’ll give another example.

**Example 3**

According to Croft et al. (2000) “Find

”

**Solution**

Let

As I can see, Here I have to integrate a product of two functions and . So I can use the method ‘integration by parts’. Now here I will choose as since is . And then I’ll be able to integrate this expression successfully. So it will be

and this gives

Next, I’ll simplify it. And that gives

As I can see, again I have to integrate a product of two functions and . Again I’ll use the integration by parts. Now I’ll choose as so that will be . And then I’ll be able to integrate this expression successfully. So it will be

and this gives

If I simplify it, I’ll get

Now I’ll integrate it further to get

And here is the integration constant.

Next, I’ll simplify it to get

So this means

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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