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THE METHOD OF INTEGRATION BY PARTS


All of the following problems use the method of integration by parts. This method uses the fact that the differential of function

$ u = f(x) $

is

$ du = f'(x) dx $ .

For example, if

$ u = x^2 - 3x + 7 $ ,

then the differential of $ u $ is

$ du = f'(x) dx = (2x -3) dx $ .

Of course, we are free to use different letters for variables. For example, if

$ w = u^3 - u $ ,

then the differential of $ w $ is

$ dw = f'(u) du = (3u^2 -1) du $ .

When working with the method of integration by parts, the differential of a function will be given first, and the function from which it came must be determined. For example, if the differential is

$ du = 2x \ dx $ ,

then the function

$ u = x^2 $

leads to the correct differential. In general, function

$ u = x^2 + C $ ,

where $ C $ is any real constant, leads to the correct differential

$ du = 2x \ dx $ .

When using the method of integration by parts, for convenience we will always choose $ C = 0 $ when determining a function (We are really finding an antiderivative when we do this.) from a given differential. For example, if the differential of $ u $ is

$ du = (x^2 + 4x -5) dx $

then the constant $ C $ can be "ignored" and the function (antiderivative) $ u $ can be chosen to be

$ u = \displaystyle{ { x^3 \over 3 } + 2x^2 - 5x } $ .

The formula for the method of integration by parts is given by

$ \displaystyle{ { \int u \, dv } = uv - { \int v \, du } } $ .

This formula follows easily from the ordinary product rule and the method of u-substitution. Theoretically, if an integral is too "difficult" to do, applying the method of integration by parts will transform this integral (left-hand side of equation) into the difference of the product of two functions and a new ``easier" integral (right-hand side of equation). It is assumed that you are familiar with the following rules of differentiation.

We will assume knowledge of the following well-known, basic indefinite integral formulas : Most of the following problems are average. A few are challenging. Make careful and precise use of the differential notation $ dx $ and $ du $ and be careful when arithmetically and algebraically simplifying expressions.







Click HERE to return to the original list of various types of calculus problems.


Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address :

kouba@math.ucdavis.edu





Duane Kouba 2000-04-23