Integration by parts

Integration by Parts (IBP) is a powerful technique for integrating the product of two functions. It’s essentially the reverse of the product rule for differentiation. The core idea is to transform a complex integral into a simpler one by “swapping” the derivative from one function to the other.

Let’s see how it’s derived.

It starts with the product rule for derivatives:

\[\frac{d}{dt}(u(t) \cdot v(t)) = u(t) \frac{d v(t)}{dt} + v(t) \frac{d u(t)}{dt}\]

Now, let’s integrate both sides with respect to \(t\) from a \(start\) to an \(end\) point:

\[\int_{start}^{end} \frac{d}{dt}(u(t) \cdot v(t)) \,dt = \int_{start}^{end} u(t) \frac{d v(t)}{dt} \,dt + \int_{start}^{end} v(t) \frac{d u(t)}{dt} \,dt\]

By the Fundamental Theorem of Calculus, the integral of the derivative on the left side simplifies to the function evaluated at the boundaries:

\[[u(t) \cdot v(t)]_{start}^{end} = \int_{start}^{end} u(t) \frac{d v(t)}{dt} \,dt + \int_{start}^{end} v(t) \frac{d u(t)}{dt} \,dt\]

Finally, we rearrange the equation to solve for one of the integrals, which gives us the formula for Integration by Parts:

\[\int_{start}^{end} u(t) \frac{d v(t)}{dt} \,dt = [u(t) \cdot v(t)]_{start}^{end} - \int_{start}^{end} v(t) \frac{d u(t)}{dt} \,dt\]

This is often written in the more compact shorthand notation:

\[\int u \,dv = uv - \int v \,du\]