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

is

.

For example, if

,

then the differential of is

.

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

,

then the differential of is

.

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

,

then the function

leads to the correct differential. In general, function

,

where is any real constant, leads to the correct differential

.

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

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

.

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

.

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.

- a.)
- b.)
- c.)
- d.)
- e.)
- f.)

- , where is a constant
- , where is a constant

*PROBLEM 1 :*Integrate .Click HERE to see a detailed solution to problem 1.

*PROBLEM 2 :*Integrate .Click HERE to see a detailed solution to problem 2.

*PROBLEM 3 :*Integrate .Click HERE to see a detailed solution to problem 3.

*PROBLEM 4 :*Integrate .Click HERE to see a detailed solution to problem 4.

*PROBLEM 5 :*Integrate .Click HERE to see a detailed solution to problem 5.

*PROBLEM 6 :*Integrate .Click HERE to see a detailed solution to problem 6.

*PROBLEM 7 :*Integrate .Click HERE to see a detailed solution to problem 7.

*PROBLEM 8 :*Integrate .Click HERE to see a detailed solution to problem 8.

*PROBLEM 9 :*Integrate .Click HERE to see a detailed solution to problem 9.

*PROBLEM 10 :*Integrate .Click HERE to see a detailed solution to problem 10.

*PROBLEM 11 :*Integrate .Click HERE to see a detailed solution to problem 11.

*PROBLEM 12 :*Integrate .Click HERE to see a detailed solution to problem 12.

*PROBLEM 13 :*Integrate .Click HERE to see a detailed solution to problem 13.

*PROBLEM 14 :*Integrate .Click HERE to see a detailed solution to problem 14.

*PROBLEM 15 :*Integrate .Click HERE to see a detailed solution to problem 15.

*PROBLEM 16 :*Integrate .Click HERE to see a detailed solution to problem 16.

*PROBLEM 17 :*Integrate .Click HERE to see a detailed solution to problem 17.

*PROBLEM 18 :*Integrate .Click HERE to see a detailed solution to problem 18.

*PROBLEM 19 :*Integrate .Click HERE to see a detailed solution to problem 19.

*PROBLEM 20 :*Integrate .Click HERE to see a detailed solution to problem 20.

*PROBLEM 21 :*Integrate .Click HERE to see a detailed solution to problem 21.

*PROBLEM 22 :*Integrate .Click HERE to see a detailed solution to problem 22.

*PROBLEM 23 :*Integrate .Click HERE to see a detailed solution to problem 23.

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

Duane Kouba 2000-04-23