13.2PureCore

Indefinite Integrals

When you integrate without limits you get an indefinite integral — and a mysterious “+ c”. That constant of integration matters: differentiating wipes out constants, so integrating must allow for any of them.

25 min Video by Zeeshan Zamurred Integration

Edexcel AS Level Maths: 13.2 Indefinite IntegralsWatch the full walkthrough before the notes below.

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What you'll be able to do

Integrate expressions term by term
Always include the constant of integration
Explain why + c is needed
Integrate functions written in awkward forms

Term by term, plus c

Integrate each term using the power rule and add a single constant of integration $c$ at the end.

1Integrate each:

x^{3}

x^{2}

- 5 x

2Add

+ c

Answer

x^{3} + x^{2} - 5 x + c

Why “+ c”?

Differentiating $x^{2} + 1$ , $x^{2} + 7$ and $x^{2} - 3$ all give $2 x$ — the constant vanishes. So integrating $2 x$ could give $x^{2}$ plus $any$ constant. We write $+ c$ to represent every possibility.

\int 2 x d x = x^{2} + c

The

+ c

stands for the unknown constant lost on differentiating.

Tip — Forgetting + c on an indefinite integral is the single most common (and costly) integration slip.

Prepare awkward integrands

As with differentiation, rewrite roots and fractions as powers, and expand brackets, before integrating.

1Rewrite as

\int x^{- 2} d x

2Raise/divide:

\frac{x ^{- 1}}{- 1}

Answer

- x^{- 1} + c = - \frac{1}{x} + c

Formula recap

\int f (x) d x = F (x) + c

Indefinite integral with + c.

\int (u + v) d x = \int u d x + \int v d x

Integrate term by term.

\int x^{- 2} d x = - x^{- 1} + c

Rewrite to power form first.

Common mistakes to avoid

Forgetting the + c.

Every indefinite integral needs + c.

Integrating 1/x² without rewriting it.

Write it as x⁻² first, then apply the rule.

Key takeaways

Integrate term by term and add a single + c.
+ c is needed because differentiation destroys constants.
Rewrite roots/fractions as powers before integrating.

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