CalculusCalculus 2College

Integrals

Antiderivatives, substitution and the Fundamental Theorem.

Integration runs differentiation in reverse and also measures accumulation. An antiderivative $F$ of $f$ satisfies $F' = f$ , and the indefinite integral $\int f(x)\,dx = F(x) + C$ records the whole family, since constants vanish when differentiated. The definite integral $\int_a^b f(x)\,dx$ is the signed area between the graph and the $x$ -axis from $a$ to $b$ .

The Fundamental Theorem of Calculus connects the two ideas: $\int_a^b f(x)\,dx = F(b) - F(a)$ . So areas, distances from velocities and totals from rates can all be computed exactly from an antiderivative, without adding up thin rectangles by hand.

Key terms

Fundamental Theorem of Calculus: A definite integral equals the change in any antiderivative.
$\int_a^b f(x)\,dx = F(b) - F(a)$

Key ideas

Power rule: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \ne -1$ , and $\int \frac{1}{x}\,dx = \ln|x| + C$ .
Substitution reverses the chain rule; integration by parts reverses the product rule.
Area below the axis counts as negative in a definite integral.
Always include $+C$ on indefinite integrals.

Worked example

Evaluate $\int_1^3 (3x^2 - 2x)\,dx$ .

Antiderivative: $F(x) = x^3 - x^2$ .
$F(3) = 27 - 9 = 18$ .
$F(1) = 1 - 1 = 0$ .
Result: $18 - 0 = 18$ .

Answer: $18$

Common mistakes

Dropping $+C$ on indefinite integrals.
Computing $F(a) - F(b)$ instead of $F(b) - F(a)$ .
Forgetting to change the limits (or substitute back) after a substitution.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

Integrals

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