Integration

Indefinite and definite integrals of powers, the area under a curve, and the trapezium rule.

9 min read

Integration reverses differentiation, and a definite integral measures the area between a curve and the $x$ -axis. The two ideas are linked by the Fundamental Theorem of Calculus.

For $n \neq = - 1$ : $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + c$ . Raise the power by one and divide by the new power. The constant $c$ is needed for an indefinite integral.

Worked example

Find $\int (x + 2)^{2} d x$ .

Expand first: $\int (x^{2} + 4 x + 4) d x = \frac{x ^{3}}{3} + 2 x^{2} + 4 x + c .$

A curve with the region between it and the x-axis, from x = a to x = b, shaded to show the definite integral. — The definite integral from a to b gives the area between the curve and the x-axis.

A definite integral is evaluated between limits: $\int_{a}^{b} f (x) d x = F (b) - F (a)$ where $F^{'} (x) = f (x)$ (the Fundamental Theorem of Calculus). It gives the (signed) area between the curve and the axis.

Worked example

Find $\int_{1}^{3} 2 x d x$ .

$[x^{2}]_{1}^{3} = 9 - 1 = 8.$

Exam tip

A definite integral gives signed area: a region below the axis counts as negative. If a question asks for the total physical area and the curve crosses the axis, split the integral at the crossing point.

When a curve cannot be integrated easily, the trapezium rule estimates the area by splitting it into thin trapezia. For a curve that bends upward (convex) the rule overestimates; for one that bends downward (concave) it underestimates.

You can also solve simple differential equations of the form $\frac{d y}{d x} = f (x)$ by integrating both sides.

Omitting " $+ c$ " on an indefinite integral.
Using $\frac{x ^{n + 1}}{n + 1}$ when $n = - 1$ (the rule fails there).
Treating signed area as physical area when the curve dips below the axis.

Integration

Overview

Integrating powers

The area under a curve

The trapezium rule

Common mistakes