Partial fraction expansion. Partial fraction decomposition to evaluate integral Opens a modal. Integration using long division Opens a modal. Integration with partial fractions. Integration using trigonometric identities. Trigonometric substitution. Introduction to trigonometric substitution Opens a modal. More trig sub practice Opens a modal. Trig and u substitution together part 1 Opens a modal. Trig and u substitution together part 2 Opens a modal.
Trig substitution with tangent Opens a modal. More trig substitution with tangent Opens a modal. Long trig sub problem Opens a modal. About this unit.
Integration by parts In this section, we study an important integration technique called integration by parts. This technique can be applied to a variety of functions and particularly useful for integrands involving products of algebraic and transcendental functions. Guidelines for using integration by parts 1. Tabular method In problems involving repeated applications of integration by parts, a tabular method can be useful. Example 3. However, instead of computing du and v we put these into the following table.
We then differentiate down the column corresponding to u until we hit zero. Be careful! It can save you a fair amount of work on occasion. To find antiderivatives for these forms, we try to break them into combinations of trigonometric integrals to which we can apply power rule.
Of course, if both exponents are odd then we can use either method. With that being said most, if not all, of integrals involving products of sines and cosines in which both exponents are even can be done using one or more of the above identities to rewrite the integrand. This means that if the exponent on the secant m is even we can strip two out and then convert the remaining secants to tangents. This means that if the exponent on the tangent n is odd and we have at least one secant in the integrand we can strip out one of the tangents along with one of the secants of course.
Note that this method does require that we have at least one secant in the integral as well. If the exponent on the secant is even and the exponent on the tangent is odd, then we can use either case. Again, it will be easier to convert the term with the smallest exponent. However, the methods used to do these integrals can also be used on some quotients involving sines and cosines and quotients involving secants and tangents and hence quotients involving cosecants and cotangents.
One such example is what we have seen in 3. At this point, I am confident that readers of this module can confidently handle integrals involving trigonometric functions.
The following section introduces us to the technique of integration by trigonometric substitutions: 3. The following theorem lists the derivatives of the six trigonometric functions. Theorem 3. Many integrals involving rational expressions can be done if we first do partial fractions on the integrand.
Recall that the degree of a polynomial is the largest exponent in the polynomial. Partial fractions can only be done if the degree of the numerator is strictly less than the degree of the denominator. That is important to remember. Then for each factor in the denominator we can use the following table to determine the term s we pick up in the partial fraction decomposition.
There are several methods for determining the coefficients for each term and we will go over each of those in the following examples. Here are a couple of points that need to be made about this strategy. It is really nothing more than a general set of guidelines that will help us to identify techniques that may work.
Some integrals can be done in more than one way and so depending on the path you take through the strategy you may end up with a different technique than somebody else who also went through this strategy. Second, while the strategy is presented as a way to identify the technique that could be used on an integral also keep in mind that, for many integrals, it can also automatically exclude certain techniques as well.
When going through the strategy keep two lists in mind. After going through the strategy and the second list has only one entry then that is the technique to use. If, on the other hand, there are more than one possible technique to use we will then have to decide on which is liable to be the best for us to use.
Unfortunately, there is no way to teach which technique is the best as that usually depends upon the person and which technique they find to be the easiest. This has already been mentioned in each of the previous points, but is important enough to warrant a separate mention. Always identify all possible techniques and then go back and determine which you feel will be the easiest for you to use. For instance, a substitution may lead to using integration by parts or partial fractions integral.
Simplify the integrand, if possible.
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