Integrate exact differentials. Hello friends, today’s topic is how to integrate exact differentials. So here it goes.

**Integrate exact differentials**

Okay, suppose I have a differential .

In one of my earlier posts on multiple integrations, I have shown how to identify an exact differential.

So, let’s say is an exact differential.

Now I want to integrate it.

And the question is: how to integrate it?

Ok, so I’ll integrate twice to get the value of . First, I’ll integrate it with respect to . Again, I’ll integrate with respect to .

When I’ll integrate with respect to , I’ll integrate only the component. So it will be

And when I’ll integrate with respect to , I’ll integrate only the component. Then it will be

Now I’ll give some examples of how to integrate exact differentials.

**Solved examples of how to integrate exact differentials**

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 is the first example.

**Example 1**

According to Stroud and Booth (2011)* “Determine which is the exact differential and integrate where appropriate to determine : .”

**Solution**

So here the given differential is

First of all, I have to find out if it’s an exact differential.

**Step 1**

So, I compare it with the standard form of the differential Then I can say that

First of all, I’ll differentiate partially with respect to .

And this gives

(1)

Next, I’ll differentiate partially with respect to .

So this gives

(2)

As I can see from both equations (1) and (2) that

Hence I can conclude that the differential is an exact differential.

Now the next step is to integrate it to get the value of .

**Step 2**

As I can see, I have to integrate twice to get the value of . First, I’ll integrate it with respect to . Again, I’ll integrate with respect to .

When I’ll integrate with respect to , I’ll integrate only the component.

And when I’ll integrate with respect to , I’ll integrate only the component.

**Now** I’ll integrate with respect to . So it will be

And this gives

Now here is the integration function of .

Then I’ll simplify the value of as

(3)

**Next**, I’ll integrate with respect to .

**Step 3**

When I integrate with respect to , it gives

So means

Now here is the integration function of .

Then I’ll simplify the value of as

(4)

Next, I’ll compare equations (3) and (4). And this will be

Now if I look at both sides, I see that

So I can put back in equation (3) to get the value of as .

Hence I can conclude that is an exact differential and . And these two are the answers to this example.

Now I’ll give another example on how to integrate exact differentials.

**Example 2**

According to Stroud and Booth (2011)* “Determine which is the exact differential and integrate where appropriate to determine : .”

**Solution**

Now here the given differential is .

First of all, I have to find out whether it’s an exact differential.

But I have already done it in Example 2 of my post on how to determine the exact differential in two variables. SoI won’t repeat that part.

Also from that example, I already know that is an exact differential.

Now I’ll integrate it.

**Step 1**

Again I’ll integrate twice to get the value of . First, I’ll integrate it with respect to . Again, I’ll integrate with respect to .

When I’ll integrate with respect to , I’ll integrate only the component.

And when I’ll integrate with respect to , I’ll integrate only the component.

**Now** I’ll integrate with respect to . So it will be

And this gives

Now here is the integration function of .

Then I’ll simplify the value of as

(5)

**Next**, I’ll integrate with respect to .

**Step 2**

When I integrate with respect to , it gives

So means

Now here is the integration function of .

Then I’ll simplify the value of as

(6)

Next, I’ll compare equations (5) and (6). And this will be

Now if I look at both sides, I see that

So I can put back in equation (5) to get the value of as .

Hence I can conclude that is an exact differential and . And these two are the answers to this example.

Now I’ll give my last example on how to integrate exact differentials.

**Example 3**

According to Stroud and Booth (2011)* “Determine which is the exact differential and integrate where appropriate to determine : .”

**Solution**

So here the given differential is

First of all, I have to find out if it’s an exact differential.

**Step 1**

So, I compare it with the standard form of the differential Then I can say that

First of all, I’ll differentiate partially with respect to .

And this gives

(7)

Next, I’ll differentiate partially with respect to .

So this gives

(8)

As I can see from both equations (7) and (8) that

Hence I can conclude that the differential is not an exact differential.

So it is not possible to integrate it in the same way as mentioned above. And this is the answer to this example.

Dear friends, this is the end of today’s post on how to integrate exact differentials. 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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GREAT ARTICLE ABOUT Integrate exact differentials

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