Integration by Parts 1

The antidifferentiation technique known as Integration by Parts or Antidifferentiation by Parts is based on the formula for the Product Rule: d\left( uv \right)=udv+vdu.
Solve this equation for the second term on the right: udv=d\left( uv \right)-vdu.
Integrating this gives the formula \int{udv}=\int{d\left( uv \right)}-\int{vdu}. By the FTC the first term on the right can be simplified giving the formula for Integration by Parts:


The technique is used to find antiderivatives of expressions such as \int{x\sin \left( x \right)dx} in which there is a combination of functions that are usually of different types – here a polynomial and a trig function.

This parts of the integrand must be matched to the parts of \int{udv}. Here we make the substitutions u=x,dv=\sin \left( x \right)dx and from these we compute du=dx\text{ and }v=-\cos \left( x \right). (There is no need for the +C here; it will be included later). Making these substitutions gives

\int{x\sin \left( x \right)dx}=-x\cos \left( x \right)-\int{-\cos \left( x \right)dx}

The integral on the right is simple so we end with

\int{x\sin \left( x \right)dx}=-x\cos \left( x \right)+\sin \left( x \right)+C

As the problems get more difficult the first question students ask is which part should by u and which dv? The rules of thumb are (1) Chose u to be something that gets simpler when differentiated, and (2) chose dv to be something you can antidifferentiated or at least something that does not get more complicated when you antidifferentiate. For example, in the problem above if we were to choose u=\sin \left( x \right) and  dv=xdx the result is

\int{x\sin \left( x \right)dx}=\tfrac{{{x}^{2}}}{2}\sin \left( x \right)-\int{\tfrac{{{x}^{2}}}{2}\cos \left( x \right)dx}

This is correct, but the integral on the right is more complicated than the one we started with. When this happens, go back and start over.

For AP Calculus teachers: Note that Antidifferentiation by Parts is a BC only topic. It is something you can do in AB classes after the AP exam if you have time.


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