13.1PureCore

Integrating xⁿ

Integration is the reverse of differentiation. Where differentiating drops the power, integrating raises it — add one to the power and divide by the new power. This single rule starts the whole topic.

25 min Video by Zeeshan Zamurred Integration

Edexcel AS Level Maths: 13.1 Integrating x^nWatch the full walkthrough before the notes below.

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What you'll be able to do

Understand integration as the reverse of differentiation
Apply the rule for integrating xⁿ
Integrate terms with coefficients
Integrate negative and fractional powers

Reverse of differentiation

Differentiating $x^{3}$ gives $3 x^{2}$ ; integrating $3 x^{2}$ takes you back to $x^{3}$ . Integration undoes differentiation — it is also called finding the $antiderivative$ .

The rule

Add one to the power, then divide by the new power. (This works for every power except $n = - 1$ .)

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + c (n \neq = - 1)

Raise the power by one, divide by the new power.

1Add one to the power:

x^{5}

2Divide by 5.

Answer

\frac{x ^{5}}{5} + c

Tip — It is the exact opposite of the power rule: differentiation multiplies then drops; integration raises then divides.

Coefficients and other powers

A coefficient is carried through. Negative and fractional powers follow the same rule — just be careful with the arithmetic of $n + 1$ .

1Integrate

x^{2}

\frac{x ^{3}}{3}

2Multiply by 6:

6 \times \frac{x ^{3}}{3}

Answer

2 x^{3} + c

Formula recap

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + c

The integration rule (n ≠ −1).

\int a x^{n} d x = \frac{a x ^{n + 1}}{n + 1} + c

Coefficients carry through.

integration = reverse of differentiation

Core idea.

Common mistakes to avoid

Dropping the power like differentiation: ∫x³ dx = 3x².

Raise and divide: ∫x³ dx = x⁴/4 + c.

Dividing by the old power instead of the new one.

Divide by (n + 1), the new power.

Key takeaways

Integration reverses differentiation.
∫xⁿ dx = xⁿ⁺¹/(n+1) + c (n ≠ −1).
Coefficients are carried through.

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