>Integrals>What Are the Important Integration Rules?

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 $d x$ ).

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

\begin{array}{l} \int k d x = k x + C \\ \int x^{n} d x = \frac{1}{n + 1} x^{n + 1} + C, n \neq - 1 \\ \int \frac{1}{x} d x = \ln | x | + C, x \neq 0 \\ \int \ln k x d x = x \ln k x - x + C \\ \int e^{x} d x = e^{x} + C \\ \int e^{k x} d x = \frac{1}{k} e^{k x} + C \\ \int \cos x d x = \sin x + C \\ \int \cos k x d x = \frac{1}{k} \sin k x + C \\ \int \sin x d x = - \cos x + C \\ \int \sin k x d x = - \frac{1}{k} \cos k x + C \\ \int \tan x d x = - \ln | \cos x | + C, \\ x \neq \frac{π}{2} + n π \\ \int \tan k x d x = - \frac{1}{k} \ln | \cos k x | + C \\ \int a^{x} d x = \frac{a^{x}}{\ln a} + C \\ \int \frac{1}{k x + a} d x = \frac{1}{k} \ln | k x + a | + C \end{array}

\begin{array}{l} \int k d x = k x + C \\ \int x^{n} d x = \frac{1}{n + 1} x^{n + 1} + C, n \neq - 1 \\ \int \frac{1}{x} d x = \ln | x | + C, x \neq 0 \\ \int \ln k x d x = x \ln k x - x + C \\ \int e^{x} d x = e^{x} + C \\ \int e^{k x} d x = \frac{1}{k} e^{k x} + C \\ \int \cos x d x = \sin x + C \\ \int \cos k x d x = \frac{1}{k} \sin k x + C \\ \int \sin x d x = - \cos x + C \\ \int \sin k x d x = - \frac{1}{k} \cos k x + C \\ \int \tan x d x = - \ln | \cos x | + C, x \neq \frac{π}{2} + n \cdot π \\ \int \tan k x d x = - \frac{1}{k} \ln | \cos k x | + C \\ \int a^{x} d x = \frac{a^{x}}{\ln a} + C \\ \int \frac{1}{k x + a} d x = \frac{1}{k} \ln | k x + a | + C \end{array}

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.

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