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 sinx is cosx, so reversing that tells you an antiderivative of cosx is sinx. What could be simpler?
Actually, there is one very little twist. Again, the derivative of sinx is cosx, but the derivative of sinx + 10 is also cosx, as is the derivative of sinx plus any constant C. So, since the derivative of sinx + C is cosx, the antiderivative of cosx is sinx + 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.