Integration by a partial fraction. Dear friends, today’s topic is integration by a partial fraction. Have a look!!

**Integration by partial fraction**

**Solved examples of the integration by partial fraction**

Disclaimer: None of these examples are 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)*¹ “Integrate the following:

”

**Solution**

Now here I have to integrate the function . So let’s say

And I’ll start with the partial fraction of the function.

As I can see, here the top, that is the numerator has the degree . And the bottom, that is the denominator has the degree . So this is an expression of lower degree numerator. Therefore I’ll use the * method for the partial fraction of lower degree numerator*.

**Step 1**

First of all, I’ll factorize the denominator, that is, . And that gives

So I can say that

Now I see that I can also bring the term on the top as well like

So that gives

(1)

Next, I’ll get the partial fraction of .

**Step 2**

Since can’t be factorized anymore and there is also another term , I’ll use two different methods together. And these are the * partial fraction of irreducible expressions* and the

*.*

**partial fraction of repetitive roots**(2)

Then I’ll simplify the right-hand side of the expression to get

Next, I’ll separate the coefficients as

Now I’ll compare the coefficients and the constants on both sides of the equation.

First of all, I’ll compare the coefficient of on both sides to get

Then I’ll compare the constants on both sides to get

Next, I’ll compare the coefficients of on both sides to get

Since , I can say that

At the end, I’ll compare the coefficients of on both sides to get

Since , I can say that

Thus I get the values of as

Then I substitute these values of and in equation (2) to get

Therefore equation (1) will be

Next, I’ll integrate this function.

**Step 3**

So it will be

Now I’ll use the standard * formulas in integration* to integrate it. And that gives

where is the integration constant.

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. (2013)*, “By writing the integrand as its partial fractions find

”

##### Solution

Now here I have to integrate the function . So let’s say

And I’ll start with the partial fraction of the function.

As I can see, here the top, that is the numerator has the degree . And the bottom, that is the denominator has the degree . So this is an expression of higher degree numerator. Therefore I’ll use the * method for the partial fraction of higher degree numerator*.

**Step 1**

First of all, I’ll factorize the numerator in such a way so that I can bring the denominator part, that is, in it. So that gives

And that means

Therefore the integrand will be

Next, I’ll get the value of .

**Step 2**

Again I’ll use the standard rules in integration to get the value of . So it will be

Here is the integration constant. Now I’ll simplify it to get

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!!

## Leave a Reply