Integration by Parts

Integration by parts is used when integrating the product of two expressions . We can also sometimes use integration by parts when we want to integrate a function that cannot be split into the product of two things.

Let’s see some examples

u = xdu = dx

dv = cos x dxSdv = Scos x dxv = sin x

More Examples

u = xdu = dx

dv = ex dxSdv = Sex dxv = ex

One More !!!

u = ln xdu = 1/x dx

dv = dxS dv = S dxv = x

Repeated Integration by Parts

u = x2

du = 2x dx dv = e-x dxSdv =S e-x dxv = -e-x

u = xdu = dx

dv = e-x dxSdv =S e-x dxv = -e-x

Integration by Parts for Definite Integrals

Example

t = 1 + x2

dt = 2x dx

½ 2 I I