Step by step
The question
Question: find the integral from 0 to 2 of x times (x² + 1)².
1 Choose u
The inside: x² + 1.
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 Change the limits
The limits 0 and 2 are x values.
u needs its own: x = 0 gives u = 1.
And x = 2 gives u = 5.
4 Write it with u
x² + 1 becomes u, and x dx becomes half of du.
A number in front can wait outside the integral.
The limits are the u values.
Careful!
Not the old limits!
0 and 2 are x values, but now we work with u.
5 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.
6 Top minus bottom
Put in 5, then take away the same with 1.
Work out the powers.
So the integral is 62/3.
7 Check
Let's check without changing the limits: put x back in the antiderivative, and use the x limits 0 and 2.
A calculator gives 20.8333 minus 0.1667: 20.6667.
The same number.
Right!
Let's look back.
First: choose u.
Then: find du.
Then: change the limits.
Then: write it with u.
Then: integrate.
Then: top minus bottom.
Last: check.