Work backwards from a rate.

An indefinite integral represents a family of functions with the same derivative. Add a constant because differentiating any constant gives zero. For a power other than $x^{-1}$ , increase the exponent by one and divide by the new exponent.

For f(ax + b), reversing the chain rule introduces a factor 1/a. Velocity integrates to position; acceleration integrates to velocity. Initial conditions determine the constants. Check your antiderivative by differentiating it.

INTERACTIVE MODEL

Try it. Watch it change.

f(x)

x^{2}

FunctionUpper limit1Trapezoids8

Integral = 0.3333 · approximation = 0.3359 · total area = 0.3333

Trapezoids approximate the signed integral. Total area adds the magnitudes of regions above and below the axis.

Explore: Switch the displayed rate. Compare the changing accumulated area with the function values.

Power rule for integration

\int x^n\,\mathrm dx=\frac{x^{n+1}}{n+1}+C

Exception

n\ne-1

WORKED EXAMPLE

Find $s(t)$ if $s'(t)$ = 6t + 2 and $s(0)$ = 5.

Integrate: $s(t)$ = $3t^{2}$ + 2t + C.
At t = 0, C = 5.
Thus $s(t)$ = $3t^{2}$ + 2t + 5.

Assumed knowledge

Differentiation and substitution.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Find antiderivatives of elementary functions.
Use an initial condition to determine the constant.

QCAA Mathematical Methods 2025 v1.3 · p. 25 ↗

READY TO TRY IT?

Work backwards from a rate.

Find s(t)s(t)s(t) if s′(t)s'(t)s′(t) = 6t + 2 and s(0)s(0)s(0) = 5.

Put the idea to work.

Find $s(t)$ if $s'(t)$ = 6t + 2 and $s(0)$ = 5.