Integration by Parts

Integration by parts is a technique of integration useful when the integrand is a product of two functions. Essentially it helps us infer the antiderivative of a function by “undoing” the product rule of differentiation:

\( \displaystyle \int {\color{DarkGoldenRod}{u}}\, {\color{DarkGreen}{v'}} \,\mathrm{d}x = {\color{DarkGoldenRod}{u}}\,{\color{DarkGreen}{v}} - \int {\color{DarkGreen}{v}}\, {\color{DarkGoldenRod}{u'}} \,\mathrm{d}x \)

Examples

\(\displaystyle \int x\cos(x) \,\mathrm{d}x \;\;=\;\; \int {\color{DarkGoldenRod}{x}}\, {\color{DarkGreen}{\cos(x)}} \,\mathrm{d}x \;\;=\;\; {\color{DarkGoldenRod}{x}}\, {\color{DarkGreen}{\sin(x)}} - \int {\color{DarkGreen}{\sin(x)}}\,{\color{DarkGoldenRod}{1}} \,\mathrm{d}x \;\;=\;\; x\sin(x) + \cos(x) + C \)

Letting \(u = x\) and \(v' = \cos(x)\) we have \(u' = 1\) and \(v = \sin(x).\)

\(\displaystyle \int \ln(x) \,\mathrm{d}x \;\;=\;\; \int {\color{DarkGoldenRod}{\ln(x)}}\, {\color{DarkGreen}{1}} \,\mathrm{d}x \;\;=\;\; {\color{DarkGoldenRod}{\ln(x)}}\, {\color{DarkGreen}{x}} - \int {\color{DarkGreen}{x}}\,{\color{DarkGoldenRod}{\tfrac{1}{x}}} \,\mathrm{d}x \;\;=\;\; {\color{DarkGoldenRod}{\ln(x)}}\, {\color{DarkGreen}{x}} - \int 1 \,\mathrm{d}x \;\;=\;\; x\ln(x) - x + C \)

Letting \(u = \ln(x)\) and \(v' = 1\) we have \(u' = \tfrac{1}{x}\) and \(v = x.\) This one is worth memorizing since \(\int\ln(x)\,\mathrm{d}x\) is so fundamental.

For a definite integral, the \(uv\) term must be evaluated at the bounds of integration: \(\int_a^b {\color{DarkGoldenRod}{u}}\, {\color{DarkGreen}{v'}} \,\mathrm{d}x = {\color{DarkGoldenRod}{u}}\,{\color{DarkGreen}{v}}\bigr\rvert_a^b - \int_a^b {\color{DarkGreen}{v}}\, {\color{DarkGoldenRod}{u'}} \,\mathrm{d}x\,. \)