Integration by Parts

This article is part of the MathHelp Tutoring Wiki

The general formula to do integration by parts:

∫u*dv=uv-∫v*du or ∫f(x)g'(x)dx=f(x)g(x)-∫g(x)f'(x)dx

EXAMPLE 1: Find the integral of xsin(x)

Step 1. Define f(x), g'(x), f'(x), g(x)

f(x)=x

g'(x)=sin(x)

f'(x)=1

g(x)=-cos(x)

Then using the formula, we can integrate.

∫xsin(x)dx=-xcos(x)-∫-cos(x)dx

= -xcos(x)+∫ cos(x)dx = -xcos(x)+ sin(x)+C **Don't forget the constant!**

EXAMPLE 2:

Find the integral of ∫x²ln(x)dx

Step 1. Define f(x), g'(x), f'(x), g(x)

f(x)=ln(x)

g'(x)=x²

f'(x)=1/x

g(x)=1/3 x³

∫ln(x)*x² dx= 1/3 ln(x)x³-∫ 1/x *1/3 x³ dx

Then using the formula, we can integrate.

= 1/3 ln(x)x³ - 1/12 ln(x) x⁴+C ***Don't forget Constant***