Integration and Area

Antiderivatives: reversing differentiation

Integration undoes differentiation. If $F'(x) = f(x)$, then $F(x)$ is an antiderivative of $f(x)$, written $\int f(x)\,dx = F(x) + C$. The constant of integration $C$ appears because any constant vanishes when differentiated, so the antiderivative is only known up to a constant.

The rule for powers reverses the power rule — raise the index by one and divide: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1.$$

Worked example. $\int (6x^2 - 4x + 5)\,dx = 2x^3 - 2x^2 + 5x + C$. You can always check integration by differentiating the answer: $\frac{d}{dx}(2x^3 - 2x^2 + 5x + C) = 6x^2 - 4x + 5$. ✓

Finding $C$ from a condition

If you know a point on the curve, you can pin down $C$. Suppose the marginal cost of production is $C'(x) = 3x^2 + 20