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10.3 u-substitution definite integralsap calculus solver

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.

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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.

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.

10.3 Usubstitution Definite Integralsap Calculus

The table below summarizes the trigonometric substitutions.

How to hire your very first employee job. 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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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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Definite Integral Example on the TI-89: Raw Transcript

10.3 U-substitution Definite Integralsap Calculus Calculator

This is a video on u substitution and this one is about a definite integral
where when you actually compute the integral you are going to compute it over a range
with a lower limit and a upper limit so anytime you have a problem like I’m going
to show you here that’s the way you use my programs
you go to u substitution let’s get started here
we’re going to press 2nd alpha and put i n d e x
into the entry line of the calculator then press alpha and put the eight and the
open and closed parenthesis press enter and you’re into my menu
you have many, many things on here notice graphing by hand
and all kinds of things that are perfectly done
and wonderfully done on these programs to help you with your tests and homework
and that’s what we’re interested in is passing calculus
and never to do it again so anyways you can scroll up or down
to u substituion this is all alphebetical and this case you
want u substitution so we might press the upper cursor to go up
and to the bottom menu, which is u substituion
notice you have trig d dx integrals for trig
half angle formulas all kinds of things you need for calculus
one for sure and then I have other programs for calculus
two and three so if I press enter we’re into u substitution
and generally you press alpha and put your function in here
you have to press alpha first and then put the function in
but I’ve already done that to speed up the video so i’m going to
I always show you, this is the function we’re doing
I haven’t put the limits in yet so you have to recognize that
you are going to do the integral first and then at the end I ask you if you want
to do the range for the area or limits
now, I a

10.3 Usubstitution Definite Integralsap Calculus