Integral calculus

How do you find the area under a curve? What about the length of any curve? Is there a way to make sense out of the idea of adding infinitely many infinitely small things? Integral calculus gives us the tools to answer these questions and many more. Surprisingly, these questions are related to the derivative, and in some sense, the answer to each one is the opposite of the derivative.
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Integrals

In this topic, we are going to connect the two big ideas in Calculus: Instantaneous rate and area under a curve. We'll see that a definite integral can be thought of as an infinite sum of infinitely small things and how this connects to the derivative of a function.

Integration techniques

We know that a definite integral can represent area and we've seen how this is connected to the idea of an antiderivative through the Fundamental Theorem of Calculus. Unfortunately, integrals aren't always easy to compute. Now, we'll build out our toolkit for evaluating integrals, both definite and indefinite!

Integration applications

Let's now use our significant arsenal of integration techniques to tackles a wide variety of problems that can be solved through integration!

Sequences, series, and function approximation

Now we switch gears away from integration to talk about sequences and series. Much of calculus is about dealing with infinity, and this topic has us dancing very closely with infinity itself. Sequences are infinite lists, series are infinite sums, and there is no small amount of delicacy involved in managing these two objects.

AP Calculus practice questions

Sample questions from the A.P. Calculus AB and BC exams (both multiple choice and free answer).

Integration techniques

We know that a definite integral can represent area and we've seen how this is connected to the idea of an antiderivative through the Fundamental Theorem of Calculus. Unfortunately, integrals aren't always easy to compute. Now, we'll build out our toolkit for evaluating integrals, both definite and indefinite!
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All content in “Integration techniques”

Integration by parts

When we wanted to take the derivative of f(x)g(x) in differential calculus, we used the product rule. In this tutorial, we use the product rule to derive a powerful way to take the anti-derivative of a class of functions--integration by parts.

u-substitution

U-substitution is a must-have tool for any integrating arsenal (tools aren't normally put in arsenals, but that sounds better than toolkit). It is essentially the reverse chain rule. Why the letter "u"? Well, it could have been anything, but this is the convention. I guess why not the letter "u" :)

Integration using trigonometric identities

You will occasionally encounter integrals in life that involve products and exponents of trig functions. In this tutorial, you will see examples of using trigonometric identities to get these types of integrals into a form that you can actually integrate.

Division and partial fraction expansion

When you're trying to integrate a rational expression, the techniques of partial fraction expansion and algebraic long division can be *very* useful.