Step by step
The question
Question: find the integral of 4x times (x² − 3)⁵.
1 Choose u
The inside: x² − 3.
Call it u.
Its derivative is 2x, and 4x is outside.
A good sign: u′ is there, or nearly.
2 Find du
The power rule: du/dx = 2x.
Multiply both sides by dx: du = 2x dx.
A small change in u is 2x times a small change in x.
We have 4x dx, not 2x dx.
4x is 2 times 2x.
So 4x dx is 2 times du.
3 Write it with u
x² − 3 becomes u, and 4x dx becomes 2 times du.
A number in front can wait outside the integral.
Careful!
4x dx is not du.
It is 2 times du.
4 Integrate
Integrate: add 1 to the power, divide by the new power.
5 plus 1 is 6.
Multiply the numbers: 2 times 1/6 is 1/3.
5 Put x back
The question was in x, so the answer is in x too.
Put x² − 3 back in place of u.
So the answer is (x² − 3)⁶ over 3 + 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, 2x.
That is 4x(x² − 3)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.