Calculus II: Integration by Parts

1. Definition:

we start from the result found for derivatives related to the product of two functions:

(f(x) g(x))' = f'(x) g(x) + f(x) g'(x)

Taking the integral of the two sides, we obtain:

f(x) g(x) = ∫f'(x) g(x) dx + ∫f(x) g'(x) dx

Therefore

∫ g(x) f'(x) dx = f(x) g(x) - ∫f(x) g'(x) dx

2. Examples

2.1. Example 1

∫ ln(x) dx = ∫ ln(x) 1.dx

f'(x) = 1 , so f(x) = x + cst
g(x) = ln(x), so ln'(x) = 1/x + cst

Therefore

∫ ln(x) dx = x ln(x) - ∫x (1/x)dx =
x ln(x) - x + cst

2.2. Example 2

∫ x exp{x} dx

dv = exp{x} dx → v = exp{x}
u = x → du = dx

∫ u dv = uv - ∫ vdu

∫ x exp{x} dx = x exp{x} - ∫ exp{x} dx =

x exp{x} - exp{x} = exp{x}(x - 1) + cst