Step by step
The question
Question: find the integral of 2x times (x² + 1)³.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Choose u
The inside: x² + 1.
Call it u.
Its derivative is 2x, and 2x 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.
3 Write it with u
x² + 1 becomes u, and 2x dx becomes du.
4 Integrate
Integrate: add 1 to the power, divide by the new power.
3 plus 1 is 4.
5 Put x back
The question was in x, so the answer is in x too.
Put x² + 1 back in place of u.
So the answer is (x² + 1)⁴ over 4 + 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 2x(x² + 1)³: the question again.
Right!
7 Pick the choice
Our answer is (x² + 1)⁴ over 4 + C.
That is choice C.
8 Why not the others
A is (x² + 1)⁴ + C.
Wrong: did not divide by 4.
B is x²(x² + 1)⁴ over 4 + C.
Wrong: kept the 2x: it is the du.
D is 3(x² + 1)² + C.
Wrong: took the power down: integrals raise it.
Let's look back.
First: choose u.
Then: find du.
Then: write it with u.
Then: integrate.
Then: put x back.
Then: check.
Then: pick the choice.
Last: why not the others.