Indefinite Integrals

An indefinite integral $\int f(x)\,dx = F(x) + C$ finds all antiderivatives of $f$ —functions whose derivative equals $f$ . The constant $C$ is mandatory because derivatives of constants are zero, so infinitely many functions share the same derivative.

Indefinite Integrals: Finding All Antiderivatives

An antiderivative of a function $f$ is a function $F$ where $F'(x) = f(x)$ . Integration reverses differentiation: instead of asking "what is the derivative?", we ask "what function has this derivative?"

The Constant of Integration

A critical insight: if $F(x) = x^2$ is an antiderivative of $2x$ , then so are $x^2 + 5$ , $x^2 - 100$ , and $x^2 + \pi$ . Since the derivative of any constant is zero, infinitely many functions differ only by a constant yet share the same derivative. This is why the general antiderivative must include an arbitrary constant:

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

The " $+C$ " is mandatory—omitting it describes only one solution from infinitely many valid answers.

Core Integration Rules

Each differentiation rule reverses into an integration rule:

Rule	Formula
Power rule	$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ ( $n \neq -1$ )
Reciprocal	$\int \frac{1}{x}\,dx = \ln\lvert x\rvert + C$
Exponential	$\int e^x\,dx = e^x + C$
Sine	$\int \sin x\,dx = -\cos x + C$
Cosine	$\int \cos x\,dx = \sin x + C$
Secant squared	$\int \sec^2 x\,dx = \tan x + C$

Notice the power rule fails at $n = -1$ (division by zero)—this is why $1/x$ requires the logarithm rule instead.

Finding Specific Antiderivatives

If you're given an initial condition like $F(0) = 3$ , use it to find the specific value of $C$ . For example, if $F'(x) = \cos x$ and $F(0) = 3$ :

$F(x) = \sin x + C$ $\sin(0) + C = 3 \implies C = 3$ $F(x) = \sin x + 3$

Verification

Always check your work by differentiating your answer. If $\frac{d}{dx}[\text{your answer}] = \text{original function}$ , you're correct. This simple habit catches most errors.

Indefinite Integrals