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.

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* = *k* *dx*.

For example,

### 3. Common Integration By Parts

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

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

### 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* = *b* *dx*) 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 = (2*x*) / 3. Draw a right triangle with angle θ, opposite side 2*x*, 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(2*x*/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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