Worked examples
The question
Question: find the integral of x eˣ.
∫ xex dx =
1 Choose u and dv
Choose u = x: its derivative is simpler, just a number.
dv is the rest: eˣ dx, which we can integrate.
u = x, dv = ex dx
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.
du = 1 dx, v = ex
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 dv = uv − ∫ v du
u times v, minus the integral of v times du.
= xex − ∫ ex dx
4 The new integral
The new integral is easy: just a number times e to a power.
= xex − ex
Careful!
It is u v minus the integral of v du.
Not plus!
Careful: not uv + ∫ v du ✗
Open the bracket: a minus in front of a bracket changes the sign of every part inside.
Add C.
So the answer is x eˣ − eˣ + C.
= xex − ex + 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.
derivative: ex + xex − ex
The other terms cancel, and we get x eˣ: the question again.
Right!
= xex
Let's look back.
First: choose u and dv.
Then: find du and v.
Then: use the rule.
Then: the new integral.
Last: check.
▶ Watch it solved
The question
Question: find the integral of x sin(2x).
∫ x sin(2x) dx =
1 Choose u and dv
Choose u = x: its derivative is simpler, just a number.
dv is the rest: sin(2x) dx, which we can integrate.
u = x, dv = sin(2x) dx
2 Find du and v
du: the derivative of u, times dx.
v: the integral of dv.
sin gives minus cos, over 2.
du = 1 dx, v = −12 cos(2x)
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 dv = uv − ∫ v du
u times v, minus the integral of v times du.
= −12x cos(2x) − ∫ (−12 cos(2x)) dx
4 The new integral
The new integral is easy: just a number times cos.
cos integrates to sin.
= −12x cos(2x) − (−14 sin(2x))
Careful!
It is u v minus the integral of v du.
Not plus!
Careful: not uv + ∫ v du ✗
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 × x cos(2x) + 1/4 × sin(2x) + C.
= −12x cos(2x) + 14 sin(2x) + 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.
derivative: −12 cos(2x) + x sin(2x) + 12 cos(2x)
The other terms cancel, and we get x sin(2x): the question again.
Right!
= x sin(2x)
Let's look back.
First: choose u and dv.
Then: find du and v.
Then: use the rule.
Then: the new integral.
Last: check.
Answer = −12x cos(2x) + 14 sin(2x) + C
▶ Watch it solved
The question
Question: find the integral of x² eˣ.
∫ x2ex dx =
1 Choose u and dv
Choose u = x²: its derivative is simpler.
dv is the rest: eˣ dx, which we can integrate.
u = x2, dv = ex dx
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.
du = 2x dx, v = ex
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 dv = uv − ∫ v du
u times v, minus the integral of v times du.
= x2ex − ∫ 2xex dx
Careful!
It is u v minus the integral of v du.
Not plus!
Careful: not uv + ∫ v du ✗
4 Parts again
The new integral has x in it again: parts again.
Choose u = 2x: its derivative is simpler, just a number.
dv is the rest: eˣ dx, which we can integrate.
u = 2x, dv = ex dx
5 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.
du = 2 dx, v = ex
6 Use the rule
u times v, minus the integral of v times du.
∫ 2xex dx = 2xex − ∫ 2ex dx
7 The new integral
The new integral is easy: just a number times e to a power.
= 2xex − 2ex
Open the bracket.
= 2xex − 2ex
Put that back in its place.
= x2ex − (2xex − 2ex)
Open the bracket: a minus in front of a bracket changes the sign of every part inside.
Add C.
So the answer is x² eˣ − 2x eˣ + 2eˣ + C.
= x2ex − 2xex + 2ex + C
8 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.
derivative: 2xex + x2ex − 2ex − 2xex + 2ex
The other terms cancel, and we get x² eˣ: the question again.
Right!
= x2ex
Let's look back.
First: choose u and dv.
Then: find du and v.
Then: use the rule.
Then: parts again.
Then: find du and v.
Then: use the rule.
Then: the new integral.
Last: check.
Answer = x2ex − 2xex + 2ex + C
▶ Watch it solved
