Lecture17 | 1 6. Techniques of Integration In this chapter we study some technique to find integrals. 6.1. Integration by parts 6.2. Partial fraction for rational functions 6.3. Integration involving trig. multiplications 6.4. Substitution by trig. functions.

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Lecture17| 1 6. Techniques of Integration

In this chapter we study some technique to find integrals.

6.1. Integration by parts

6.2. Partial fraction for rational functions

6.3. Integration involving trig. multiplications

6.4. Substitution by trig. functions.

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Lecture17| 2 Integral formulas

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Lecture17| 3

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Lecture17| 4 6.1. Integration by Parts

The integration by parts (or simply, by parts) is often employed to get integral of

(i) a product

(ii) a non-product

It is based on the following observation.

Product rule. , so

For example, so

so

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Lecture17| 5

and , so

Consider the last example, let

Then

Thus we obtain

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Lecture17| 6

From the preceding example, we get:

Integration by parts.

and in the special case , we get

It is often written as

because and .

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Lecture17| 7 A criterion to find .

For an integral , suppose can be recognized as a derivative

and is simpler (e.g. or ),

choose

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Lecture17| 8 EXAMPLE. Find the integral

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EXAMPLE ( ).

Find the integral

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Lecture17| 13 6.2. Partial Fraction for Rational Functions

In this section we employ a basic result from

elementary math about written out any fraction into smaller partial fractions. For example,

so

Consider an integral

where are polynomials in .

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Lecture17| 14 If is not in a proper form, i.e.

has degree higher higher or equal to the degree of , we need a long division to get

For example,

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Lecture17| 16 Assume is a proper form. We need to

factorize into linear or quadratic factors.

EXAMPLE.

are two simple linear factors.

is a simple linear factor

is a repeated linear factor ( )

is a simple linear factor,

is a repeated linear factor ( ), is a quadratic factor.

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Lecture17| 17 Decomposition of partial fraction.

Let be proper. Consider the partial fraction (PF) of .

1. If is a simple linear factor for , the PF should contain

where is a number TBC.

2. If is a repeated linear factor of , the PF should contain

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where are numbers TBC.

3. If is a simple quadratic factor of , then the PF should contain

4. If is a repeated quad. factor of , then the PF should contain

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Lecture17| 19 are calculated after gathering all fractions (from the factorization of ).

For , by factorize

so

We combine RHS and multiply LHS denominator

substitute some numbers

So and

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The above calculation method works well when contains only linear factors.

Integral formulas

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