Step by step
The question
Question: find the integral of 3x² times √(x³ + 1).
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 Write it with u
x³ + 1 becomes u, and 3x² dx becomes du.
A square root is the power one half.
4 Integrate
Integrate: add 1 to the power, divide by the new power.
One half plus 1 is three halves.
Dividing by three halves is the same as times 2/3.
5 Put x back
The question was in x, so the answer is in x too.
Put x³ + 1 back in place of u.
So the answer is 2/3 × (x³ + 1)^(3/2) + 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 + 1): 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.