### 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 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.) We will assume knowledge of the following well-known, basic indefinite integral formulas :
1. , where is a constant 2. 3. 4. 5. 6. , where is a constant
7. Most of the following problems are average. A few are challenging. Make careful and precise use of the differential notation and and be careful when arithmetically and algebraically simplifying expressions.

• PROBLEM 1 : Integrate .

• PROBLEM 2 : Integrate .

• PROBLEM 3 : Integrate .

• PROBLEM 4 : Integrate .

• PROBLEM 5 : Integrate .

• PROBLEM 6 : Integrate .

• PROBLEM 7 : Integrate .

• PROBLEM 8 : Integrate .

• PROBLEM 9 : Integrate .

• PROBLEM 10 : Integrate .

• PROBLEM 11 : Integrate .

• PROBLEM 12 : Integrate .

• PROBLEM 13 : Integrate .

• PROBLEM 14 : Integrate .

• PROBLEM 15 : Integrate .

• PROBLEM 16 : Integrate .

• PROBLEM 17 : Integrate .

• PROBLEM 18 : Integrate .

• PROBLEM 19 : Integrate .

• PROBLEM 20 : Integrate .

• PROBLEM 21 : Integrate .

• PROBLEM 22 : Integrate .

• PROBLEM 23 : Integrate .