Step by step
The question
Question: find the integral of x times (x² + 5)².
1 Choose u
The inside: x² + 5.
Call it u.
Its derivative is 2x, and x 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 x dx, not 2x dx.
x is half of 2x.
So x dx is half of du.
3 Write it with u
x² + 5 becomes u, and x dx becomes half of du.
A number in front can wait outside the integral.
Careful!
x dx is not du.
It is only half of du.
4 Integrate
Integrate: add 1 to the power, divide by the new power.
2 plus 1 is 3.
Multiply the numbers: 1/2 times 1/3 is 1/6.
5 Put x back
The question was in x, so the answer is in x too.
Put x² + 5 back in place of u.
So the answer is (x² + 5)³ over 6 + 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 x(x² + 5)²: 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.