1. 8.2 Integration by Parts

2. Summary of Common Integrals Using
Integration by Parts
1. For integrals of the form
Let u = xn and let dv = eax dx, sin ax dx, cos ax dx
2. For integrals of the form
Let u = lnx, arcsin ax, or arctan ax and let dv = xn dx
3. For integrals of the form or
Let u = sin bx or cos bx and let dv = eax dx

3. Integration by Parts
If u and v are functions of x and have
continuous derivatives, then

4. Guidelines for Integration by Parts
Try letting dv be the most complicated
portion of the integrand that fits a basic
integration formula. Then u will be the
remaining factor(s) of the integrand.
Try letting u be the portion of the
integrand whose derivative is a simpler
function than u. Then dv will be the
remaining factor(s) of the integrand.

5. Evaluate u dv u dv u dv u dv To apply integration by parts, we want
to write the integral in the form
There are several ways to do
this.
u dv u dv u dv u dv
Following our guidelines, we choose the first option
because the derivative of u = x is the simplest and
dv = ex dx is the most complicated.

6. u = x
v = ex
du = dx
dv = ex dx
u dv

7. Since x2 integrates easier than
ln x, let u = ln x and dv = x2
u = ln x
dv = x2 dx

8. Repeated application of integration by parts
u = x2
v = -cos x
du = 2x dx
dv = sin x dx
Apply integration by
parts again.
u = 2x
du = 2 dx
dv = cos x dx
v = sin x

9. Repeated application of integration by parts
Neither of these choices for u
differentiate down to nothing,
so we can let u = ex or sin x.
Let’s let u = sin x
v = ex
du = cos x dx
dv = ex dx
u = cos x
v = ex
du = -sinx dx
dv = ex dx