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

Read more on similar topics like:

**Reduction formulas for algebraic functions**

**Integrate Sec^n (x)**

**Solved example of how to i**ntegrate Sec^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)*, “If , prove that

Hence evaluate

”

**Solution**

Now here the integral is

And I have to prove that

Also, I’ll get the value of .

So I’ll start with the integration of .

**Step 1**

As you can see, I can also rewrite as

Then I’ll use the method ‘* integration by parts*‘.

As I know, this method says

Thus, in this example, I’ll choose as and as .

So this gives

Then I’ll simplify it to get

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

Thus will be

(1)

Since , I can say that .

Therefore equation (1) becomes

Then I’ll simplify it to get

(2)

Hence I have proved the relation.

Now I’ll do the next part.

**Step 2**

So I’ll get the value of . And that means I have to get the value of with the integration limits as .

Now I’ll substitute in equation (2) to get

(3)

Then I’ll get the value of . So I’ll substitute in equation (2) to get

(4)

Now I’ll substitute in equation (2) to get

(5)

Then I’ll substitute in equation (2) to get

And I’ll simplify it. So that gives

Since , I can say that

Now I’ll simplify it further to get

(6)

**Step 3**

Next, I’ll put back the value of from equation (6) to equation (5) to get

Then I’ll simplify it to get

Now I’ll substitute this value of to equation (4) to get

If I simplify it, I get

Next, I’ll substitute this value of to equation (3) to get

Then I’ll simplify it to get

Now I’ll take out the common terms. And that means

If I simplify it further, it becomes

As a next step, I’ll find out the value of for which the limiting values of are .

** Step 4**

So that means

Since is a constant term, it can be out from the brackets as

Now I’ll substitute the limits to get

Also I already know that

Next, I’ll substitute these values above to get

Then I’ll simplify it to get

So this means

Now I’ll simplify it further to get

Thus the value of is which is correct upto significant figures.

Hence I can conclude that this is the second solution of 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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