Reduction Formula for Integral of ∫cos^n(x)dx

In this video, we work through the derivation of the reduction formula for the integral of cos^n(x) or [cos(x)]^n.

The first step is to rewrite the integral as:

∫cos^n(x)dx = ∫cos^n-1(x)cos(x)dx

Thus we have 2 parts to the integral, where:

u = cos^n-1(x)
dv = cos(x)dx

We can then proceed with integration by parts:

∫udv = uv - ∫vdu

Working through the substitutions and the steps, we eventually arrive at the reduction formula shown in the video.

Thanks for watching. Please give me a "thumbs up" if you have found this video helpful.

Please ask me a maths question by commenting below and I will try to help you in future videos.

[carlosgallegos]

when you wrote cosine to the n-1 I almost pulled all my hair off my head trying to justify it to then just letting the video go on and realize I was initially right.
It was really useful though, thank you!

[adriansguitarvirtuoso]

U deserve not just tips but wages sir !!!!.... Unfortunately I don't have that to offer for now, but just know that your efforts are really appreciated

[abdullahimaalim]

Perfect. There you have my thumb up. I like the way you did last step. Thank you.

[chebukatigodwinmk]

Wooow, sir you've reaally helped me. Thanks a lot. When the lecturer taught me this at first I was blank, but now I can even do it on my own. Regards

[rjaychigman]

l think using I(n) makes things a little bit difficult if u are a beginner. I would rather recommend using I(n+1) as your starting formula. From there you basically use the same principles and will get your answer as I(n+1) = [cos^n(x)sin(x) + nI(n-1)]/n...then substitute n with (n-1) to make the formula in terms of huh?

[biebie]

Thank you so much, you literally saved my

[danianjan]

Amazing ! Thanks very very much for this solution...

[curiosity]

Amazing Sir !
Thanks to Allah, I found your channel.

[wristdisabledwriter]

I swear I don’t remember learning this in calc 2 class. I decided to try it and check my answer using the video. I got it right. The difference was I used the di method

[srnois]

thank you so much! i was stuck on this for literal hours and you helped me out! thanks again

[kengazpaul]

that's a great explanation keep it up👏👏👏

[doaa_alshawwa]

thanks for real it was super helpfull !!

[SankitDas.]

7:51 what about integration of cos x^(n-2)dx kaise karega yaar agar n=5/8, 1/8 etc

[OneandonlyKiwi]

6:18 how did it equal cos^n, shouldn’t it just be N?

[kipkiruingetich]

Will you do one with polynomial function

[hussainmohammed]

Thank you soon much sir...its really very helpful for me...

[chin]

Demonstrate that integrate (cos^m(x) cos^n(x)) with upper limit is (pi/2), lower limit is 0 dx; = ((m/(m+n)) integrate (cos^m-1(x) cos(n-1)x with upper limit is (pi/2), lower limit is 0 dx. Hence, deduce that integrate (cos^m(x) cos(nx) dx = (pi/(2^(n+1)) with upper limit is (pi/2), lower limit is 0? how to demonstrate
it?

[technicaltrading]

If you don't mind ..can you tell me which software you are using for writing.. plz reply

[danchenqijiang]

Fantastic! Just got stuck with this integral to show the volume of n-dim volume of a unit ball by Fubini's Theorem. lol

[thomasmetias]

3:05 why u put (n-1)cosx^(n-2) equals to (n-1)cos^(n-1) ??