Step by step
The question
Question: find the integral of cos⁵ x.
1 Save one
The power of cos is 5: odd.
Keep one cos x aside.
It will be part of du.
2 Change the rest
The rule: sin squared plus cos squared is 1.
So cos² x = 1 − sin² x.
Now everything else is sin x.
3 Choose u
Let u = sin x.
Its derivative is cos x: du = cos x dx.
That is the cos x dx we kept.
4 Write it with u
sin x becomes u, and cos x dx becomes du.
Multiply out: (1 − u²)² is 1 − 2u² + u⁴.
Then times the rest.
Careful!
The power rule does not work here: the derivative of cos x would have to be there too.
Save one and swap the rest.
5 Integrate
The power rule: add 1 to each power, divide by the new power.
6 Put x back
Put sin x back in place of u.
So the answer is sin⁵ x/5 − 2sin³ x/3 + sin x + C.
7 Check
Let's check: the slope of our answer F near x = 1, with a calculator in radians: about 0.046.
The question at x = 1 gives 0.046.
The same: the slope of the answer is the question.
Right!
Let's look back.
First: save one.
Then: change the rest.
Then: choose u.
Then: write it with u.
Then: integrate.
Then: put x back.
Last: check.