Integration by Parts

In differential calculus, we derived formulas to differentiate products of functions.

Recall that

d f(x)g(x)dx=f(x)g(x)+f(x)g(x).\frac{d\ \color{teal}f(x) \color{green}g(x)}{dx} = \color{teal}f'(x)\color{green}g(x)\color{currentcolor}+\color{teal}f(x)\color{green}g'(x)\color{currentcolor}.

Integration by Parts is a method to do the same sort of thing, for integrals. The formula can be derived simply by re-arranging the product rule (shown above).

In order to see this a bit better, we'll perform the following substitution:

h(x)=f(x).\textcolor{orchid}{h(x)} = \textcolor{teal}{f'(x)}.

The Fundamental Theorem of Calculus tells us that

h(x) dx=f(x).\int \textcolor{orchid}{h(x)}\ dx = \textcolor{teal}{f(x)}.

Let's substitute! Replacing all occurrences of f(x),\textcolor{teal}{f(x)}, we get:

d h(x) dx g(x)dx=h(x)g(x)+h(x) dxg(x).\frac{d\ \int \textcolor{orchid}{h(x)}\ dx\ \cdot \textcolor{green}{g\left(x\right)}}{dx}=\textcolor{orchid}{h\left(x\right)}\textcolor{green}{g\left(x\right)}+\int \textcolor{orchid}{h(x)}\ dx\cdot \textcolor{green}{g'\left(x\right)}.

Now, there's a big ugly ddx\frac{d}{dx} here - let's get rid of it! Remember - integration kills differentiation, and vice-versa!

So, we integrate both sides.

g(x)h(x) dx=\textcolor{green}{g\left(x\right)} \int \textcolor{orchid}{h(x)}\ dx=
h(x) g(x) dx+g(x)h(x)dx dx.\int \textcolor{orchid}{h(x)}~\textcolor{green}{g\left(x\right)}\ dx+ \int\textcolor{green}{g'\left(x\right)}\int \textcolor{orchid}{h(x)}dx \ dx.

Rearranging to isolate h(x) g(x) dx\int \textcolor{orchid}{h(x)}~\textcolor{green}{g\left(x\right)}\ dx, we see that

h(x) g(x) dx=\int \textcolor{orchid}{h(x)}~\textcolor{green}{g\left(x\right)}\ dx =
g(x)h(x) dxg(x)h(x) dx dx.\textcolor{green}{g\left(x\right)} \int \textcolor{orchid}{h(x)}\ dx - \int \textcolor{green}{g'\left(x\right)} \int \textcolor{orchid}{h(x)}\ dx \ dx.

And, this is integration by parts!

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