Integrate Tan^n (x). Hello friends, today I’ll talk about how to integrate Tan^n (x). Have a look!!

Read more on similar topics like:

**Reduction formulas for algebraic functions**

**Integrate Sec^n (x) | How to integrate a function of the form Sec^n (x)**

**Integrate Tan^n (x)**

**Solved example of how to i**ntegrate Tan^n (x)

Disclaimer: This example is not mine. I have chosen it from some book. I have also given the due reference at the end of the post.

So here is my example.

**Example **

According to Stroud and Booth (2013)*, “By writing as , obtain a reduction formula for

Hence show that

”

**Solution**

Now here I have to get a reduction formula for . Also I have to prove that

Therefore I’ll start with the integration of .

**Step 1**

As I can see from the example itself, I’ll write as

Also, from the * standard formulas in trigonometry*, I know that

Therefore I can say that

So the reduction formula for will be

Then I’ll simplify it as

And that gives

Now I’ll integrate . Thus it will be

And that means

where is the integration constant. So this gives

Therefore will be

If I say, , then it will be

Also, if , then

Thus the reduction formula for is

(1)

And this is the answer to the first part of this example.

Next, I’ll show that .

**Step 2**

Thus I’ll substitute the limits in equation (1). So it will be

Also, I already know that

So will be

Then I’ll simplify it to get

Therefore I have also proved the relation in the second part as well.

Hence I can conclude that these are the solutions 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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