Strategy for Integration

in mathematics •  2 months ago  (edited)

In this video I go over a useful four-step strategy of integration. This strategy requires understanding and memorization of the basic integration formulas as well as understanding the various integration techniques. In this video I go over ways in selecting the proper integration techniques to use for any given integral and also show how sometimes the most useful method is to simply try random techniques in hopes that the solution will be more recognizable. Also, when stuck in any given integral it is important to try different techniques again as often times multiple techniques or even combination of integration techniques are required for integrating the integral.


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Strategy for Integration

Strategy of Integration.jpeg

As I have shown in my earlier videos, integration is more challenging than differentiation.

  • It is obvious which differentiation formula to apply but it is not always obvious which integration technique to use.

In my earlier videos, I have covered individual techniques for specific cases such as integration by substitution, by parts, and by partial fractions.

Selecting which technique may be challenging but understanding and memorizing the basic techniques and formulas is important in developing a strategy for integration.

The following Table of Integration Formulas is important to memorize:

  • Note: the formulas in asterisk (*) need not be memorized because they are easily derived
    • Formula 19 can be avoided by using partial fractions
    • Formula 20 can be avoided by using trigonometric substitution

Once you understand and memorize these basic integration formulas and if you don't immediately see how to solve an integral, then you can try to following four-step strategy:

Four-Step Integration Strategy

(1) Simplify the Integrand if Possible:

Sometimes using algebraic manipulation or trigonometric identities will simplify the integrand and make the method of integration obvious. Here are some examples:

(2) Look for an Obvious Substitution

Try to find some function u = g(x) in the integrand whose differential du = g'(x) also occurs, apart from a constant factor as illustrated in the following example:

(3) Classify the Integrand According to its Form

If steps 1 and 2 have not led to the solution then we take a look at the form of the integrand f(x)

(a) Trigonometric functions:

  • If f(x) is a product of powers of sin(x) and cos(x), of tan(x) and sec(x), or of cot(x) and csc(x), then we use the trigonometric substitutions

(b) Rational functions:

  • If f is a rational function, we use the method of partial fractions

(c) Integration by parts:

  • If f(x) is a product of a power of x (or a polynomial) and a transcendental function (such as a trigonometric, exponential, or logarithmic function), then we try integration by parts.
    • This uses the fact that the derivative of a polynomial gets simpler and makes the integration easer.

(d) Radicals:

Particular kinds of substitutions are recommended when certain radicals appear.

  • (i) Use trigonometric substitution if the following radical occurs:

  • (ii) Use a rationalizing substitution for the following radical:

More generally, this sometimes works for:

(4) Try Again

  • If the first three steps have not produced the answer, remember that there are basically only two methods of integration: substitution and by parts.

(a) Try substitution

  • Even if no substitution is obvious (step 2), some inspiration or ingenuity (or even desperation) may suggest an appropriate substitution

(b) Try parts

  • Although integration by parts is used most of the time on products of the form described in Step 3(c), it is sometimes effective on single functions. From my earlier videos, I have shown that it works on tan-1(x), sin-1(x), and ln(x); note that these are all inverse functions

(c) Manipulate the integrand

  • Algebraic manipulations (perhaps rationalizing the denominator or using trigonometric identities) may be useful in transforming the integral into an easier form. These manipulations may be more substantial than in Step 1 and may involve some ingenuity. Here is an example:

(d) Relate the problem to previous problems

  • When you have built up some experience in integration, you may be able to use a method or a given integral that is similar to a method you have already used on a previous integral.
  • Or you may even be able to express the given integral in terms of a previous one as in the following example:

(e) Use several methods

  • Sometimes two or three methods are required to evaluate an integral.
  • The evaluation could involve several successive substitutions of different types, or it might combine integration by parts with one or more substitutions, etc.

In later videos I will go over some examples illustrating the process of choosing appropriate methods.

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