Step by step
The question
Question: find the integral from 0 to π/2 of cos³ x.
1 Save one
The power of cos is 3: 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 the bracket.
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.
No + C: top minus bottom.
7 Top minus bottom
The values of sin at the limits.
Put them in: top minus bottom.
So the integral is 2/3.
8 Check
Let's check with a calculator: it adds up 1000 thin strips.
That gives about 0.667, the same as our answer.
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.
Then: top minus bottom.
Last: check.