Integration by parts

For two continuously differentiable functions F(x) and G(x), by the product rule:

\displaystyle\frac{d}{dx}\left[F(x)G(x)\right] = F(x)\frac{dG}{dx} + G(x)\frac{dF}{dx}

Take the integral of both sides between x=a and x=b:

\displaystyle\int_a^b \frac{d}{dx}\left[F(x)G(x)\,dx = \displaystyle\int_a^b F(x)\frac{dG}{dx}\,dx + \displaystyle\int_a^b G(x)\frac{dF}{dx}\,dx

Apply the fundamental theorem of calculus to the left side of the equation. We are left with:

\eval{F(x)G(x)}_a^b = \displaystyle\int_a^b F(x)\frac{dG}{dx}\,dx + \displaystyle\int_a^b G(x)\frac{dF}{dx}\,dx

A form of the above equation that is more often used is:

\displaystyle\int_a^b F(x)\frac{dG}{dx}\,dx=\eval{F(x)G(x)}_a^b - \displaystyle\int_a^b G(x)\frac{dF}{dx}\,dx