7 17 2 Test Prepa Calculus Pdf
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7 17 2 Test Prepa Calculus Answers

The AP Calculus exams include a substantial amount of integration. So it’s very important to be familiar with integrals, numerous integration methods, and the interpretations and applications of integration. In this short article, we’ll take a look at some of the most common integrals on the test.

For a quick review of integration (or, antidifferentiation), you might want to check out the following articles first.

Use our free AP Calculus AB multiple choice to prepare for your exam. We have 10 unit tests which cover the major topics of this course, followed by a full-length AP Calculus AB practice exam. Answers and detailed explanations are included with all of our practice questions. Choose a test from the listing below to start your AP Calc review. If you plan to take Calculus 1, and are currently in Precalculus, wait to take the placement exam until you have completed at least 90% of the course. Directions for Taking the Precalculus and Calculus 1 Math Placement Exam. Make sure you are ready to take the exam (see policies above). You may take the placement test only once per testing period.

And now, without further ado, here are some of the most common integrals found on the AP Calculus exams!

Common Integrals

The following seven integrals (or their close cousins) seem to pop up all the time on the AP Calculus AB and BC exams.

1. Remember your Trig Integrals!

Trigonometric functions are popular on the exam!

2. Simple Substitutions

You need to recognize when to use the substitution u = kx, for constant k. This substitution generates a factor of 1/k because du = kdx.

For example,

3. Common Integration By Parts

Integrands of the form xf(x) often lend themselves to integration by parts (IBP).

In the following integral, let u = x and dv = sin xdx, and use IBP.

7 17 2 Test Prepa Calculus Pdf

4. Linear Denominators

Integrands of the form a/(bx + c) pop up as a result of partial fractions decomposition. (See AP Calculus BC Review: Partial Fractions). While partial fractions is a BC test topic, it’s not rare to see an integral with linear denominator showing up in the AB test as well.

The key is that substituting u = bx + c (and du = bdx) turns the integrand into a constant times 1/u. Let’s see how this works in general. Keep in mind that a, b, and c must be constants in order to use this rule.

5. Integral of Ln x

The antiderivative of f(x) = ln x is interesting. You have to use a tricky integration by parts.

Let u = ln x, and dv = dx.

By the way, this trick works for other inverse functions too, such as the inverse trig functions, arcsin x, arccos x, and arctan x. For example,

6. Using Trig Identities

For some trigonometric integrals, you have to rewrite the integrand in an equivalent way. In other words, use a trig identity before integrating. One of the most popular (and useful) techniques is the half-angle identity.

7. Trigonometric Substitution

It’s no secret that the AP Calculus exams consist of challenging problems. Perhaps the most challenging integrals are those that require a trigonometric substitution.

The table below summarizes the trigonometric substitutions.

For example, find the integral:

Here, the best substitution would be x = (3/2) sin θ.

Now we’re not out of the woods yet. Use the half-angle identity (see point 6 above). We also get to use the double-angle identity for sine in the second line.

Note, the third line may seem like it comes out of nowhere. But it’s based on the substitution and a right triangle.

If x = (3/2) sin θ, then sin θ = (2x) / 3. Draw a right triangle with angle θ, opposite side 2x, and hypotenuse 3.

By the Pythagorean Theorem, we find the adjacent side is equal to:

That allows us to identify cos θ in the expression (adjacent over hypotenuse).

Finally, θ by itself is equal to arcsin(2x/3).

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About Shaun Ault

Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!

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Your decision to take the AP Calculus BC exam involves many factors, but in essence it boils down to a question of choosing between three hours of time spent on the AP Calculus BC exam and a significant amount of time spent in a college classroom. Depending on the college, a score of 3, 4, or 5 on the AP Calculus exam can allow you to leap over the introductory courses and jump right into more advanced classes. These advanced classes are usually smaller, more specifically focused, more intellectually stimulating, and simply more interesting than a basic course. If you are concerned solely about fulfilling your mathematics requirement so you can get on with your study of pre-Columbian art, Elizabethan music, or some other area apart from calculus, the AP exam can help you there, too.

How are the AP Calculus AB and BC exams different?

AP Calculus BC is an extension of AP Calculus AB, and as such the difference between the two is in scope, not difficulty. Both AP Calculus AB and BC are full-year high school courses roughly equivalent to first-year college calculus classes. AP Calculus BC includes topics from AP Calculus AB, such as integrals and the fundamental theorem of calculus, plus other concepts such as parametric equations, polar and vector functions and series.

What topics are tested on the AP Calculus BC exam?

Unit	Topic	Percentage
Unit 1	Limits and Continuity	4–7%
Unit 2	Differentiation: Definition and Fundamental Properties	4–7%
Unit 3	Differentiation: Composite, Implicit, and Inverse Functions	4–7%
Unit 4	Contextual Applications of Differentiation	6–9%
Unit 5	Analytical Applications of Differentiation	8–11%
Unit 6	Integration and Accumulation of Change

7.17.2 Test Prepap Calculus