If you're seeing this message, it means we're having trouble loading external resources for Khan Academy.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Integral calculus

Would you believe me if I told you that if you walked straight at a wall that you would never actually get to the wall? Integral calculus allows you to mathematically prove this crazy idea. When you think of calculus, think tiny as in infinitesimal. By subdividing the space between you and the wall into ever smaller divisions, you can mathematically establish that there is an infinite number of divisions, and you can never actually get to the wall. Do not try this at home kids, not without some help from integrals and derivatives, the basic tools of calculus. The study of integral calculus includes: integrals and their inverse, differentials, derivatives, anti-derivatives, and approximating the area of curvilinear regions.
Community Questions
Integration techniques
We know that a definite integral can represent area and we've seen how this is connected to the idea of an anti-derivative through the Fundamental Theorem of Calculus (which is why we also use the integration symbol for anti-derivatives as well). Now, we'll build out our toolkit for evaluating integrals, both definite and indefinite!
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 reverise chain rule. U-substitution is very useful for any integral where an expression is of the form g(f(x))f'(x)(and a few other cases). Over time, you'll be able to do these in your head without necessarily even explicitly substituting. Why the letter "u"? Well, it could have been anything, but this is the convention. I guess why not the letter "u" :)

Reverse chain rule

The Chain Rule tells us that derivative of g(f(x)) = g'(f(x))f'(x). You already knew this. But what about going the other way around? What happens if you want to integrate g'(f(x))f'(x)? Well, that's what the "reverse chain rule" is for. As you can see, a lot of integrals you'll run into can be solved this way. It is also another way of doing u-substitution without having to substitute (so it is faster)!

Integration using trigonometric identities

You will occasionally encounter integrals in life that involve products of 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 (and you'll get some practice doing it as well)!

Trigonometric substitution

We will now do another substitution technique (the other was u-substitution) where we substitute variables with trig functions. This allows us to leverage some trigonometric identities to simplify the expression into one that it is easier to take the anti-derivative of.

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.