Integration by Parts

• Method of Substitution– Related to the chain rule

• Integration by Parts– Related to the product rule– More complex to implement than the Method

of Substitution

Derivation of Integration by Parts Formula

Let u and v be differentiable functions of x.

duvvudvu

duvdvuvu

dxuvdxvuvu

dxuvvudxvu

uvvuvu

Derivation of Integration by Parts Formula

Let u and v be differentiable functions of x.

(Product Rule)

duvvudvu

duvdvuvu

dxuvdxvuvu

dxuvvudxvu

uvvuvu

Derivation of Integration by Parts Formula

Let u and v be differentiable functions of x.

(Product Rule)

(Integrate both sides)

duvvudvu

duvdvuvu

dxuvdxvuvu

dxuvvudxvu

uvvuvu

Derivation of Integration by Parts Formula

Let u and v be differentiable functions of x.

(Product Rule)

(Integrate both sides)

(FTC; sum rule)

duvvudvu

duvdvuvu

dxuvdxvuvu

dxuvvudxvu

uvvuvu

Derivation of Integration by Parts Formula

Let u and v be differentiable functions of x.

(Product Rule)

(Integrate both sides)

(FTC; sum rule)

dx

duu

dx

dvv ,

Derivation of Integration by Parts Formula

Let u and v be differentiable functions of x.

(Product Rule)

(Integrate both sides)

(FTC; sum rule)

dx

duu

dx

dvv ,

(Rearrange terms)

duvvudvu

duvdvuvu

dxuvdxvuvu

dxuvvudxvu

uvvuvu

Integration by Parts Formula

• What good does it do us?– We can trade one integral for another.– This is only helpful if the integral we start with

is difficult and we can trade it for a good (i.e., solvable) one.

duvvudvu

b

a

b

a

b

a

duvvudvu

Classic Example

dxex x

Helpful Hints

• For u, choose a function whose derivative is “nicer”.– LIATE

• dv must include everything else (including dx).