Integration by parts is a technique of integration which states $\int uv' = uv - \int vu'$ When using integration by parts, a good tip is to let $v$ be the part for which it is easiest to find the [[antiderivative]]. ## Derivation This can be shown from the [[product rule]], which recall is $(u*v)' = uv' + vu'$ If we integrate both sides we get $uv = \int uv' + \int vu'$ Finally, rearranging terms we have the integration by parts formula $\int uv' = uv - \int vu'$ ## With limits When provided limits of integration, you need to evaluate both terms for the limits. $\int_a^b uv' = uv\Big |_a^b - \int_a^bvu'$ ## Example Let's solve the following integral $\int x^2 ln(x) dx$ The term $x^2$ will be easiest to find the antiderivative of and so we'll let $u = ln(x)$ and $v' = x^2 dx$. Finding the antiderivative of $x^2dx$ we get $v=\frac13x^3$Finding the derivative of $ln(x)$ we get $u' = \frac1xdx$ Plugging into the integration by parts formula, we have $\int x^2 ln(x) dx = \frac13x^3ln(x) - \int\frac13x^3\frac1xdx$ Simplifying we have $\int x^2 ln(x) dx = \frac13 x^3 ln(x)-\frac13 \int x^2 dx$ and solving for the integral $\int x^2 ln(x) dx = \frac13 x^3 ln(x) - \frac13 * \frac13x^3 + C$ which can be simplified further to $\int x^2 ln(x) dx = \frac13 x^3 ln(x) - \frac19 x^3 + C$ If the limits of integration were provided we can drop the constant and use the [[Fundamental Theorem of Calculus]] by plugging the upper and lower limits into the equation and calculating the difference.