Step by step
The question
Question: find the integral of (2x − 1)⁵.
1 Choose u
The inside: 2x − 1.
Call it u.
Its derivative is just a number, 2.
2 Find du
The power rule: du/dx = 2.
Multiply both sides by dx: du = 2 dx.
A small change in u is 2 times a small change in x.
We have just dx.
Divide both sides by 2: dx = du over 2.
3 Write it with u
2x − 1 becomes u, and dx becomes half of du.
A number in front can wait outside the integral.
Careful!
dx is not du.
It is only half of du.
4 Integrate
Integrate: add 1 to the power, divide by the new power.
5 plus 1 is 6.
Multiply the numbers: 1/2 times 1/6 is 1/12.
5 Put x back
The question was in x, so the answer is in x too.
Put 2x − 1 back in place of u.
So the answer is (2x − 1)⁶ over 12 + 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, 2.
That is (2x − 1)⁵: 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.