Step by step
The question
Question: find the integral of x ln x.
1 Choose u and dv
Choose u = ln x: its derivative, 1/x, is simpler.
We do not know how to integrate ln x yet.
So dv is the rest.
2 Find du and v
The derivative of ln x is 1 over x.
And v is the integral of dv: 1/2 × x², by the power rule.
3 Use the rule
The rule: the integral of u dv is u v, minus the integral of v du.
It comes from the product rule, backwards.
u times v, minus the integral of v times du.
4 The new integral
The new integral is easy: the power rule.
Careful!
It is u v minus the integral of v du.
Not plus!
Open the bracket: a minus in front of a bracket changes the sign of every part inside.
Add C.
So the answer is 1/2 × x² ln x − 1/4 × x² + C.
5 Check
Let's check: take the derivative of our answer.
The product rule on each term: the first's slope times the second, plus the first times the second's slope.
The other terms cancel, and we get x ln x: the question again.
Right!
Let's look back.
First: choose u and dv.
Then: find du and v.
Then: use the rule.
Then: the new integral.
Last: check.