Integration summary

Key learning points

• $\int ax^n\,dx=\frac{a}{n+1}x^{n+1}+C$ ($n\ne-1$).
• Definite integral: substitute upper minus lower limit.
• Area between curve and axis uses definite integrals (watch signs below the axis).
• Check by differentiating your answer.

Worked example 1 — Indefinite integral

Find $\int(6x^2-4)\,dx$.

Working

Include $+C$ for indefinite integrals.

Worked example 2 — ()

Use this method from the Integration sheet: $\int ax^n\,dx=\frac{a}{n+1}x^{n+1}+C$ ($n\ne-1$).

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 3 — Definite integral: substitute upper minus lower limit

Use this method from the Integration sheet: Definite integral: substitute upper minus lower limit.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 4 — Area between curve and axis uses definite integrals

Use this method from the Integration sheet: Area between curve and axis uses definite integrals (watch signs below the axis).

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 5 — Check by differentiating your answer

Use this method from the Integration sheet: Check by differentiating your answer.

Working

Each key learning point matches a question type on the printable worksheet.

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.