Integrate Cosh^(n) x dx. Hello friends, today I’ll show how to integrate any function of the form Cosh^(n) x dx. Have a look!!

Want to know more about the reduction formulas of other functions?? Have a look at the followings:

**Reduction formulas for trigonometric functions**

**Reduction formula for Sin^(n) x | Integrate Sin^(n) x dx****Get the reduction formula for Cos^(n) x | Integrate Cos^(n) x dx****Reduction formula for Tan^(n) x | Integrate Tan^(n) x dx****Integrate Cot^(n) x dx | Reduction formula for Cot^(n) x****Reduction formula for Sec^(n) x | Integrate Sec^(n) x dx**

**Reduction formulas for logarithmic functions**

**Reduction formulas for algebraic functions**

**DOWNLOAD, READ & PRINT – Integrate Cosh^(n) x dx**

**Integrate Cosh^(n) x dx**

Now here I’ll show how to integrate a function of the form . Also, in other words, I can say that I’ll get the reduction formula for the hyperbolic function .

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

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 , prove that

Hence evaluate

**Solution**

As I can see, I have to get the value of where

As I can see, it’s not possible to integrate straight away. So I’ll try some other way.

**Step 1**

Thus I can write as . So will be

Now I’ll use the * integration by parts* method. And that says

So in this case, I’ll choose as and as . If , then If , then

Thus becomes

Next, I’ll simplify it to get

Also, I can rewrite as

Thus becomes

Then I’ll simplify it. And that gives

(1)

Now comes my next step.

**Step 2**

As I can see, I can’t integrate further. If , then will be

Thus equation (1) becomes

Now I’ll simplify it to get

So I can say that

Hence I have proved the relation. And this is the answer to the first part of the given example.

Next, I’ll get the value of

**Step 3**

First of all, I’ll get the value of . Then I’ll substitute the limits. So here . Thus it will be

(2)

Now for , becomes . So the value of is

where is the integration constant.

Next, I’ll substitute in equation (2) to get

Then I’ll put the limits to get the value of

**Step 4**

As I can see, here is the upper limit and is the lower limit of integration.

So becomes

Since and , I can say that

Now I’ll substitute to get the value of as

Next, I’ll simplify it. Since , I can say that . Thus will be

(3)

Now I’ll get the value of .

**Step 5**

As I have already mentioned in Step 1, I can rewrite as

So this means

Next, I’ll replace with . And that gives

Also, I have already mentioned in Step 4, . Thus the value of will be

Now I’ll simplify it to get

Then I’ll substitute in equation (3) to get the value of as

Hence I can conclude that this is the solution of the last part of this 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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