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Antiderivative

An antiderivative is a function that has the given function as its derivative. You always add a constant, because the constant disappears when you differentiate. The power rule increases the exponent by one and divides by the new exponent. It does not work for $1/x$ , where the answer is the natural logarithm. For $e^{kx}$ you divide by the coefficient in the exponent.

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C

the power rule

\int e^{kx}\,dx = \frac{1}{k}e^{kx} + C

exponential

Symbols

$C$	the constant of integration
$n$	the exponent

Example

$\int 3x^2\,dx = x^3 + C$ . At $x = 2$ , without the constant, the value is $8$ . $\int e^{2x}\,dx = \dfrac{1}{2}e^{2x} + C$ , and from $0$ to $1$ the increase is $\approx 3.195$ .

Check an antiderivative by differentiating it. You should get back the function you integrated.

Practise integration for free →

The definite integral →

Part of Mathematics R2: Integration.