Tabular Integration by Parts

To understand where the tabular method for integration by parts comes from, start with the basic parts formula:

Apply parts to the integral on the right, differentiating du/dx and integrating v. This gives:

If I apply parts again to the integral on the right, I obtain

I could keep going indefinitely --- there's nothing special about doing it 3 times --- but this is enough to see the pattern.

What's going on? Look at the terms on the right side of the equation.

The signs of the terms alternate (+, -, +, -, and so on). Each term is a derivative of u times an integral of v --- except the last term, which is the integral of a derivative of u times an integral of v. The pattern is captured in the following table:

• The first column consists of alternating +'s and -'s.
• The second column consists of the successive derivatives of u.
• The third column consists of the successive intgrals of v. (You omit the arbitrary constants.)
• You obtain the terms in the parts formula by multiplying the terms in the shaded areas.

For example, look at the equation before the table. The first term on the right is uv; I get that from the table by multiplying the "+", the "u", and the "v" in the first shaded area. Check for yourself that the next two terms in the equation come from the second and third shaded areas in the table.

An obvious question is: How do you stop? The table keeps going downward, because you can take derivatives and integrals forever.

In some cases, the derivatives become zero after a while, and the table stops automatically. You'll see this in one of the examples.

In general, if you want to stop the table after a certain point, you integrate all the terms on a given row:

Compare the shaded area in the table above to the last term in the parts equation:

They're the same!

With all the symbols flying around, you may have found this explanation rough going. Don't worry --- look at the examples and you'll see that in practice this approach is really easy.

• Introduction: Integration by parts
• How do you set up an integration by parts table?
• The integral of x^2 e^x
  • How do you choose what to put in the table?
• The integral of ln x
• The integral of e^(2x) cos (3x)
• A recursion formula for the integral of (sin x)^n
• Computing a definite integral using integration by parts

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Last updated: June 13, 2005

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