Step by step
The question
Question: find the integral of (5x + 1)³.
1 Choose u
The inside: 5x + 1.
Call it u.
Its derivative is just a number, 5.
2 Find du
The power rule: du/dx = 5.
Multiply both sides by dx: du = 5 dx.
A small change in u is 5 times a small change in x.
We have just dx.
Divide both sides by 5: dx = du over 5.
3 Write it with u
5x + 1 becomes u, and dx becomes a fifth of du.
A number in front can wait outside the integral.
Careful!
dx is not du.
It is only a fifth of du.
4 Integrate
Integrate: add 1 to the power, divide by the new power.
3 plus 1 is 4.
Multiply the numbers: 1/5 times 1/4 is 1/20.
5 Put x back
The question was in x, so the answer is in x too.
Put 5x + 1 back in place of u.
So the answer is (5x + 1)⁴ over 20 + 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, 5.
That is (5x + 1)³: 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.