The integral of e^(2x) cos (3x)

To use parts to compute the integral of e^(2x) cos (3x), I have to divide the stuff in the integral up into two chunks. I'll let u = e^(2x) and dv = cos(3x) dx. (You could also follow the L-I-P-T-E rule and use u = cos (3x) and dv = e^(2x) dx; this is a case where it doesn't matter.

After three rows, things seem to be going in circles:

Notice that the integral on the right side is the same as the original integral! It may seem as though things have gone terribly wrong.

The key is to stop working on the right side of the equation, and remember that what I have is an equation. Since I'm trying to find the integral of e^(2x) cos (3x), I'll solve the equation for it by moving the copy on the right to the left: Just add "(4/9)(the integral)" to both sides:

The "13/9" is 1 + 4/9: that is, 1 times the original integral added to 4/9 times the original integral.

To solve for the original integral, I multiplied both sides of the equation by 9/13 to clear the fractions.

It often happens that applying parts produces a copy of the original integral. As long as the copy does not have the same coefficient (+1) as the original copy, you can solve for the original integral using the trick above.

• 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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