Colorful House of Math logo
Log in

What Are the Important Integration Rules?


Integration is to derivation as a plus is to a minus. That means that just like a minus reverses what a plus does, and vice versa, integration reverses what derivation does, and vice versa. Another way of putting it is that integration is finding the anti-derivative.

Now that you know this, you have also learned a method to check that your integrals are correct: After integrating, you can differentiate your answer to see whether you get the original expression that you integrated (called the integrand, the expression between the integral sign and dx).

Below is an overview of the integrals of several known and important functions. You will be expected to know these integral rules by heart. I recommend that you memorize them to learn them well, and that you always write down the proper formula before applying it when solving exercises.

Formula

Integration Formulas

kdx = kx + C xndx = 1 n + 1xn+1 + C,n 1 1 xdx = ln |x| + C,x0 ln kxdx = x ln kx x + C exdx = ex + C ekxdx = 1 kekx + C cos xdx = sin x + C cos kxdx = 1 k sin kx + C sin xdx = cos x + C sin kxdx = 1 k cos kx + C tan xdx = ln |cos x| + C, tan xdx = xπ 2 + nπ tan kxdx = 1 k ln |cos kx| + C axdx = ax ln a + C 1 kx + adx = 1 k ln |kx + a| + C

kdx = kx + C xndx = 1 n + 1xn+1 + C,n 1 1 xdx = ln |x| + C,x0 ln kxdx = x ln kx x + C exdx = ex + C ekxdx = 1 kekx + C cos xdx = sin x + C cos kxdx = 1 k sin kx + C sin xdx = cos x + C sin kxdx = 1 k cos kx + C tan xdx = ln |cos x| + C,xπ 2 + n π tan kxdx = 1 k ln |cos kx| + C axdx = ax ln a + C 1 kx + adx = 1 k ln |kx + a| + C

Note! The constant C is not as mysterious as it might seem. It represents a possible constant that disappears when you differentiate. Because we don’t know what the constant is, C can represent any number.

Want to know more?Sign UpIt's free!
White arrow pointing to the LeftPrevious entry
What Are Riemann Sums Used for?
Next entryWhite arrow pointing to the right
Which Basic Relations in Integration Should You Know?