{"id":8788,"date":"2017-01-30T17:06:49","date_gmt":"2017-01-31T01:06:49","guid":{"rendered":"https:\/\/magoosh.com\/hs\/?p=8788"},"modified":"2018-10-24T03:40:09","modified_gmt":"2018-10-24T10:40:09","slug":"ap-calculus-exam-review-antidifferentiation","status":"publish","type":"post","link":"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-exam-review-antidifferentiation\/","title":{"rendered":"AP Calculus Exam Review: Antidifferentiation"},"content":{"rendered":"

Antidifferentiation is an essential tool for evaluating integrals. In fact, you might say that antidifferentiation is pretty much the same thing as finding an indefinite integral<\/a>.<\/p>\n

When we antidifferentiate a function, we are simply finding another function whose derivative is equal to the one we started with.<\/p>\n

What is Antidifferentiation?<\/h2>\n

Antidifferentiation<\/strong> is the reverse of differentiation<\/strong>. So just as differentiation is the process of finding a derivative, anti<\/em>differentiation is the catch-all term for any formula, technique, or method of finding an anti<\/em>derivative.<\/p>\n

If f<\/em> is a function, then we say that F<\/em> is an antiderivative<\/strong> for f<\/em> if F<\/em> '(x<\/em>) = f<\/em>(x<\/em>).<\/p>\n

How Many Antiderivatives Can a Function Have?<\/h3>\n

If a function has one antiderivative, then it has infinitely<\/em> many antiderivatives. This is because the derivative of any constant is 0. So if F<\/em>(x<\/em>) is an antiderivative of f<\/em>(x<\/em>) and C<\/em> is any constant, then<\/p>\n

[F<\/em>(x<\/em>) + C<\/em>]' = F<\/em> '(x<\/em>) + [C]' = f<\/em>(x<\/em>) + 0 = f<\/em>(x<\/em>).<\/p>\n

Therefore, F<\/em>(x<\/em>) + C<\/em> is also<\/em> an antiderivative for f<\/em>. We call C<\/em> the constant of integration<\/strong>.<\/p>\n

Notation and Relationship to Integral<\/h3>\n

The indefinite integral<\/strong> of a function f<\/em> is, by definition, the most general antiderivative of f<\/em>. Notation for the indefinite integral is shown below.<\/p>\n

\"Indefinite<\/p>\n

Whenever we work out an indefinite integral, we are doing antidifferentiation.<\/p>\n

Antidifferentiation Formulas<\/h2>\n

There are many more formulas and techniques for antidifferentiation. Fortunately you only need to be aware of the basics for both the AB and BC exams.<\/p>\n

The AP Calculus AB exam covers basic antidifferentiation formulas and substitution of variables.<\/p>\n

The AP Calculus BC exam includes not only the basic formulas and substitution but also integration by parts (IBP), and simple partial fractions (PF). This test also has more challenging problems overall.<\/p>\n

Note, these rules are all written in terms of indefinite integrals.<\/p>\n

The Basic Rules<\/h2>\n

The “basic” rules refer to those rules that follow directly from differentiation rules. Some of the more common rules include the Power Rule, Sum\/Difference Rule, and Constant Multiple Rule. There are also rules for certain trigonometric, exponential, and other elementary functions.<\/p>\n

\"Power<\/p>\n

\"Sum<\/p>\n

\"Constant<\/p>\n

\"Constant<\/p>\n

\"Rule<\/p>\n

\"Exponential<\/p>\n

\"Trigonometric<\/p>\n

Substitution Rule<\/h3>\n

The Substitution Rule<\/strong>, or u<\/em>-substitution, is a rule that “reverses” the Chain Rule.<\/p>\n

\"Substitution<\/p>\n

Integration by Parts<\/h3>\n

Integration by Parts<\/strong> (IBP) is a powerful method that may be used when there are certain kinds of products in the integrand. In fact, you can think of IBP as a way to “reverse” the Product Rule.<\/p>\n

Suppose u<\/em> and v<\/em> are differentiable functions of x<\/em>. Then the IBP formula states that:<\/p>\n

\"Integration<\/p>\n

Partial Fractions<\/h3>\n

The method of Partial Fractions<\/strong> (PF) is an specialized technique for rational functions<\/em>. The main idea is to break apart the fraction into a sum of\u00a0simpler fractions. <\/p>\n

Fortunately, even on the BC exam, you only need to know the easiest cases of PF. In the simplest cases, the denominator factors into unique linear<\/em> factors. <\/p>\n