Integration by Parts 2

Sometimes when doing an Antidifferentiation by Parts, the resulting integral is simpler than the one you started with but requires another, perhaps several more, antidifferentiations. You can do this but it can get a little complicated keeping track of everything especially with all the minus signs. There is an easier way.

Let’s consider an example: \int{{{x}^{4}}\sin \left( x \right)dx}.

Begin by making a table as shown below. After the headings:

    • In the first column leave the first cell blank and then alternate plus and minus signs down the column.
    • In the second column leave the first cell blank and then enter u and then under it list its successive derivatives .
    • In the third column enter dv in the first cell and then list its successive antiderivatives under it

Tabular 5

The antiderivative is found by multiplying across each row starting with the row with the  first plus sign and adding the products:

\int{{{x}^{4}}\sin \left( x \right)dx}=

-{{x}^{4}}\cos \left( x \right)+4{{x}^{3}}\sin \left( x \right)+12{{x}^{2}}\cos \left( x \right)-24x\sin \left( x \right)-24\cos \left( x \right)+C

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