# How to Find Antiderivatives with Reverse Rules

The easiest antiderivative rules are the ones that are simply the reverse of derivative rules that you probably already know. These rules are automatic, one-step antiderivatives, with the exception of the reverse power rule, which is only slightly harder.

You know that the derivative of sin*x* is cos*x,* so reversing that tells you an antiderivative of cos*x* is sin*x**.* What could be simpler?

Actually, there is one very little twist. Again, the derivative of sin*x* is cos*x*, but the derivative of sin*x* + 10 is also cos*x*, as is the derivative of sin*x* plus any constant *C*. So, since the derivative of sin*x* + *C* is cos*x*, the antiderivative of cos*x* is sin*x* + *C*. In symbols, you write

Here’s a list of the reverse rules for antiderivatives.

By the power rule, you know that

Bring the power in front where it will

*multiply*the rest of the derivative.*Reduce*the power by one and simplify.

To reverse this process, you reverse the order of the two steps and reverse the math within each step. Here’s how that works with this example:

*Increase*the power by one.The 3 becomes a 4.

*Divide*by the new power and simplify.

Especially when you’re new to antidifferentiation, it’s a good idea to test your antiderivatives by differentiating them — you can ignore the *C.* If you get back to your original function, you know your antiderivative is correct.