Integration 1 Topic Review summary

Key learning points

• Reverse differentiation systematically; check by differentiating.
• Standard forms for $e^{kx}$, $\sin kx$, $\cos kx$ with chain-rule factors.
• Definite integrals and area between a curve and the $x$-axis.
• Split integrals at roots when a curve crosses the axis.
• Substitution when the inner derivative is visible.

Worked example 1 — Standard

$\int e^{3x}\,dx$.

Working

Chain-rule factor 3.

Worked example 2 — Area

Area under $y=\sqrt{x}$ from 1 to 4.

Working

$F=\frac23 x^{3/2}$.

Worked example 3 — Reverse differentiation systematically; ch

Apply this idea from the Topic Review: Reverse differentiation systematically; check by differentiating.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Standard forms for e^{kx}, \\sin kx, \\cos

Standard forms for $e^{kx}$, $\\sin kx$, $\\cos kx$ with chain-rule factors.

Working

Attempt without notes, then check the answers PDF.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.