THE METHOD OF INTEGRATION BY PARTS



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

[pic]

is

[pic].

For example, if

[pic],

then the differential of [pic]is

[pic].

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

[pic],

then the differential of [pic]is

[pic].

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

[pic],

then the function

[pic]

leads to the correct differential. In general, function

[pic],

where [pic]is any real constant, leads to the correct differential

[pic].

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

[pic]

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

[pic].

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

[pic].

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.

o a.) [pic]

o b.) [pic]

o c.) [pic]

o d.) [pic]

o e.) [pic]

o f.) [pic]

We will assume knowledge of the following well-known, basic indefinite integral formulas :

1. [pic], where [pic]is a constant [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic], where [pic]is a constant

7. [pic]

Most of the following problems are average. A few are challenging. Make careful and precise use of the differential notation [pic]and [pic]and be careful when arithmetically and algebraically simplifying expressions

SOLUTIONS TO INTEGRATION BY PARTS

SOLUTION 1 : Integrate [pic]. Let

[pic]and [pic]

so that

[pic]and [pic].

Therefore,

[pic]

[pic].

SOLUTION 2 : Integrate [pic]. Let

[pic]and [pic]

so that

[pic]and [pic].

Therefore,

[pic]

[pic]

[pic].

SOLUTION 3 : Integrate [pic]. Let

[pic]and [pic]

so that

[pic]and [pic].

Therefore,

[pic]

[pic]

[pic]

[pic].

SOLUTION 4 : Integrate [pic]. Let

[pic]and [pic]

so that

[pic]and [pic].

Therefore,

[pic]

[pic]

[pic]

[pic].

SOLUTION 5 : Integrate [pic]. Let

[pic]and [pic]

so that

[pic]and [pic].

Therefore,

[pic]

[pic]

[pic]

[pic]

[pic].

SOLUTION 6 : Integrate [pic]. Let

[pic]and [pic]

so that (Don't forget to use the chain rule when differentiating [pic].)

[pic]and [pic].

Therefore,

[pic]

[pic].

Now use u-substitution. Let

[pic]

so that

[pic],

or

[pic].

Then

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]+ C

[pic]+ C

[pic]+ C .

SOLUTION 7 : Integrate [pic]. Let

[pic]and [pic]

so that

[pic]and [pic].

Therefore,

[pic]

[pic]

[pic].

SOLUTION 8 : Integrate [pic]. Let

[pic]and [pic]

so that

[pic]and [pic].

Therefore,

[pic]

(Add [pic]in the numerator. This will replicate the denominator and allow us to split the function into two parts.)

[pic]

[pic]

[pic]

[pic].

SOLUTION 9 : Integrate [pic]. Let

[pic]and [pic]

so that

[pic]and [pic].

Therefore,

[pic]

[pic].

Integrate by parts again. Let

[pic]and [pic]

so that

[pic]and [pic].

Hence,

[pic]

[pic]

[pic]

[pic]

[pic].

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