How to integrate dx/(acos x + bsin x + c). Dear friends, today I’ll show how to integrate dx/(acos x + bsin x + c). Have a look!!

Want to know more about the integration of different forms of expressions? Check these out:

**dx/(a cos^2x+bsin^2x+c) – how to integrate it?**

integrate_2_compressed

### Integrate dx/(acos x + bsin x + c)

Now I’ll integrate

So let’s say . Then will be

And will be

Then will be

Also, will be

If I differentiate , I get

So this means

Since , will be

Thus I can write in terms of and then integrate it accordingly.

Now I’ll give you some examples of that.

**Solved examples of how to integrate dx/(a cos x + b sin x + c)**

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 (2013)*, “Evaluate the following:

”

**Solution**

Now here I have to evaluate the expression, say as

First of all, I’ll replace with as I have mentioned above.

**Step 1**

So let’s say . Then will be

And will be

Then will be

(Please check above for detailed work.)

Also, will be

Similarly, will be

Thus will be

Then I’ll simplify it to get

Since , I can say that

Now I’ll use the same technique as how to integrate dz/(a^2 – z^2).

**Step 2**

First of all, I’ll write it in the form of

So will be

Next, I’ll rewrite it as

Then I’ll simplify it. Thus it will be

Therefore I can say

Now I’ll use the following formula to get the value of as

So, in this example, it will be

And here is the integration constant.

Next, I’ll simplify it to get the value of as

Then I’ll replace with to get the value of . Thus it will be

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

Now I’ll give you another example.

**Example 2**

According to Stroud and Booth (2013)*, “Evaluate the following:

”

**Solution**

Now here I have to evaluate the expression, say as

And I can also rewrite it as

So this gives the value of as

Thus I can say that

Now let’s say that

(1)

Next, I’ll find out the value of .

First of all, I’ll replace with in as I have mentioned above in example 1.

**Step 1**

So let’s say . Then will be

And will be

Then will be

(Please check above for detailed work.)

Also, will be

Similarly, will be

Thus will be

Then I’ll simplify it to get

Since , I can say that

Next, I’ll take out from as

As I can see, I can use the same method as the **integration of **.

**Step 2**

So that gives

And is the integration constant.

Now I’ll simplify it to get

Then I’ll replace with to get

Next, I’ll put back this value of to equation (1) as

Thus it becomes

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

Now I’ll give you another example.

**Example 3**

According to Stroud and Booth (2013)*, “Evaluate the following:

”

**Solution**

Now here I have to evaluate the expression, say as

First of all, I’ll replace with as I have mentioned above.

**Step 1**

So let’s say . Then will be

And will be

Then will be

(Please check above for detailed work.)

Also, will be

Similarly, will be

Thus will be

Then I’ll simplify it to get

Since , I can say that

Now I’ll integrate it.

**Step 2**

Since has the form as , I’ll use the same method as in example 2.

So it will be

And is the integration constant.

Then I’ll replace with to get the value of . Thus it will be

Hence I can conclude that this is the solution 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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