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
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 integrate 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.
According to Stroud and Booth (2013)*, “By writing , prove that
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.
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
Next, I’ll simplify it to get
Also, I can rewrite as
Then I’ll simplify it. And that gives
Now comes my next step.
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
First of all, I’ll get the value of . Then I’ll substitute the limits. So here . Thus it will be
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
As I can see, here is the upper limit and is the lower limit of integration.
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
Now I’ll get the value of .
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!!