Step by step
The question
Question: find the integral of 2x eˣ.
1 Choose u and dv
Choose u = 2x: its derivative is simpler, just a number.
dv is the rest: eˣ dx, which we can integrate.
2 Find du and v
du: the derivative of u, times dx.
v: the integral of dv.
e to the x stays e to the x.
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: just a number times e to a power.
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 2x eˣ − 2eˣ + 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 2x eˣ: 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.