Step by step
The question
Question: find the integral from 1 to 2 of 3x² times (x³ − 1)².
1 Choose u
The inside: x³ − 1.
Call it u.
Its derivative is 3x², and 3x² is outside.
A good sign: u′ is there, or nearly.
2 Find du
The power rule: du/dx = 3x².
Multiply both sides by dx: du = 3x² dx.
A small change in u is 3x² times a small change in x.
3 Change the limits
The limits 1 and 2 are x values.
u needs its own: x = 1 gives u = 0.
And x = 2 gives u = 7.
4 Write it with u
x³ − 1 becomes u, and 3x² dx becomes du.
The limits are the u values.
Careful!
Not the old limits!
1 and 2 are x values, but now we work with u.
5 Integrate
Integrate: add 1 to the power, divide by the new power.
2 plus 1 is 3.
6 Top minus bottom
Put in 7, then take away the same with 0.
Work out the powers.
So the integral is 343/3.
7 Check
Let's check without changing the limits: put x back in the antiderivative, and use the x limits 1 and 2.
A calculator gives 114.3333 minus 0: 114.3333.
The same number.
Right!
Let's look back.
First: choose u.
Then: find du.
Then: change the limits.
Then: write it with u.
Then: integrate.
Then: top minus bottom.
Last: check.