Integrals (Antiderivatives)

What is an integral?

An integral undoes a derivative. If $F'(x) = f(x)$, then $F$ is an antiderivative of $f$, and we write:

$\int f(x)\, dx = F(x) + C$

The $+ C$ is the constant of integration: any constant differentiates to zero, so antiderivatives are only unique up to a constant.

The reverse power rule

Differentiation gives $\frac{d}{dx} x^n = n x^{n-1}$. Reversing it:

$\int x^n\, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$

Examples.

• $\int x^3\, dx = \frac{x^4}{4} + C$
• $\int x\, dx = \frac{x^2}{2} + C$ (with $n = 1$)
• $\int 1\, dx = x + C$ (with $n = 0$)
• $\int \frac{1}{x}\, dx = \ln|x| + C$ (the special $n = -1$ case)

Constants and sums

$\int k \cdot f(x)\, dx = k \int f(x)\, dx$ $\int [f(x) + g(x)]\, dx = \int f(x)\, dx + \int g(x)\, dx$

Example. $\int (6x^2 - 4x + 5)\, dx = 2x^3 - 2x^2 + 5x + C$.

Common integrals

$f(x)$	$\int f(x)\, dx$
$\cos x$	$\sin x + C$
$\sin x$	$-\cos x + C$
$e^x$	$e^x + C$
$1/x$	$\ln \lvert x \rvert + C$
$\sec^2 x$	$\tan x + C$

Definite integrals

A definite integral has bounds:

$\int_a^b f(x)\, dx = F(b) - F(a)$

This is the Fundamental Theorem of Calculus — integrals and derivatives are inverse operations. Geometrically, the definite integral equals the (signed) area under $f$ between $x = a$ and $x = b$.

Example. $\int_0^2 3x^2\, dx$. Antiderivative is $x^3$. Evaluate: $2^3 - 0^3 = 8$.

Key takeaways

• An integral reverses a derivative; add $+\, C$ for indefinite integrals.
• Reverse power rule: $\int x^n\, dx = \frac{x^{n+1}}{n+1} + C$ (when $n \neq -1$).
• Definite integrals use bounds: evaluate the antiderivative at the top, subtract at the bottom.
• Area under the curve = definite integral.