Step by step
The question
Question: find the integral of 3x² times (x³ − 2)⁴.
1 Choose u
The inside: x³ − 2.
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 Write it with u
x³ − 2 becomes u, and 3x² dx becomes du.
4 Integrate
Integrate: add 1 to the power, divide by the new power.
4 plus 1 is 5.
5 Put x back
The question was in x, so the answer is in x too.
Put x³ − 2 back in place of u.
So the answer is (x³ − 2)⁵ over 5 + C.
6 Check
Let's check: take the derivative of our answer with the chain rule.
The power comes down and cancels the number below.
Then times the inside's slope, 3x².
That is 3x²(x³ − 2)⁴: the question again.
Right!
Let's look back.
First: choose u.
Then: find du.
Then: write it with u.
Then: integrate.
Then: put x back.
Last: check.