IB Maths AA SL Topic 5 — Calculus Paper 1 & 2 ~7 min read

Integrating Powers of x

There’s one rule for integrating any power of x: raise the power by 1, then divide by the new power. It’s literally the power rule running backwards. Memorise it and you can integrate any polynomial in seconds.

📘 What you need to know

The power rule: ∫ xⁿ dx = x^{n + 1}n + 1 + c (in formula booklet, valid for n ≠ −1).
With a constant in front: ∫ axⁿ dx = ax^{n + 1}n + 1 + c.
Constant special case: ∫ a dx = ax + c.
Works for any rational power except n = −1 (the rule fails — you’d divide by zero).
Rewrite roots and fractions as powers of x first.
Integrate term by term for sums and differences. Expand products before integrating.
Don’t forget “+ c” for indefinite integrals.

The power rule for integration

Power rule for integration

∫ xⁿ dx = x^{n + 1}n + 1 + c (n ≠ −1)

✓ in formula booklet

In words: add 1 to the power, then divide by the new power. Watch it work on x⁴:

x⁴power = 4 → x⁵/5 + cadd 1 → 5 · divide by 5

“Raise the power by 1, divide by the new power”

🧠

“Raise & divide”

Two steps every time. Raise the power by 1. Divide by that new power. Add + c. (Compare this with differentiation’s “drop & drop” — they’re literal opposites.)

Special cases

Constants

∫ a dx = ax + c

e.g. ∫ 4 dx = 4x + c

⚠ The exception

∫ x⁻¹ dx ≠ formula!

power rule fails for n = −1 (you’d divide by 0)

The constant rule looks weird, but it’s just the power rule with n = 0: ∫ ax⁰ dx = ax¹/1 + c = ax + c. The exception n = −1 needs a different rule (involves ln) — covered later in the course.

Rewrite first, integrate second

Same trick as differentiation — get every term into the form axⁿ before applying the rule.

Original form	Rewrite as	Why
√x	x^1/2	square root = power of ½
³√x	x^1/3	cube root = power of ⅓
4x²	4x⁻²	denominator’s power becomes negative
1√x	x^−1/2	root in denominator → negative fraction

Sums, differences, and products

Sums & differences → integrate term by term · Products → expand brackets first

📍

Don’t try to integrate products directly

For ∫ 8x²(2x − 3) dx, expand first to get 16x³ − 24x², then integrate term by term. There’s no “product rule for integration” at AA SL.

Worked examples

WE 1

Apply the power rule

Find: (a) ∫ x⁵ dx · (b) ∫ 4x³ dx · (c) ∫ 6 dx

Raise & divide. Don’t forget +c.part (a) ∫ x⁵ dx = x⁶/6 + cpart (b) ∫ 4x³ dx = 4x⁴/4 + c = x⁴ + cpart (c) ∫ 6 dx = 6x + c x⁶/6 + c · x⁴ + c · 6x + c notice (b): the 4 cancels nicely!

WE 2

Sums and differences — term by term

Find ∫ (8x³ − 2x + 4) dx.

8x³ → 8x⁴/4 = 2x⁴ −2x → −2x²/2 = −x² 4 → 4x ∫ (8x³ − 2x + 4) dx = 2x⁴ − x² + 4x + c one “+c” for the whole answer — not one per term!

WE 3

Roots and fractions — rewrite first

Given that dy/dx = 3x⁴ − 2x² + 3 − 1/√x, find an expression for y in terms of x.

step 1 — rewrite 1/√x = x^(−1/2): dy/dx = 3x⁴ − 2x² + 3 − x^(−1/2)step 2 — integrate term by term 3x⁴ → 3x⁵/5 −2x² → −2x³/3 3 → 3x −x^(−1/2) → −x^(1/2)/(1/2) = −2x^(1/2) = −2√xy = (3/5)x⁵ − (2/3)x³ + 3x − 2√x + c −½ + 1 = ½ — careful with negative fractional powers!

WE 4

Products — expand first

Find ∫ 8x²(2x − 3) dx.

step 1 — expand 8x²(2x − 3) = 16x³ − 24x²step 2 — integrate term by term 16x³ → 16x⁴/4 = 4x⁴ −24x² → −24x³/3 = −8x³ ∫ 8x²(2x − 3) dx = 4x⁴ − 8x³ + c always expand brackets BEFORE integrating!

WE 5

Mixed — roots, negative powers

Find ∫ (5√x + 2/x³) dx.

step 1 — rewrite 5√x = 5x^(1/2) 2/x³ = 2x^(−3) ∫ (5x^(1/2) + 2x^(−3)) dxstep 2 — integrate 5x^(1/2) → 5x^(3/2)/(3/2) = (10/3)x^(3/2) 2x^(−3) → 2x^(−2)/(−2) = −x^(−2) ∫ = (10/3)x^(3/2) − x^(−2) + c divide by a fraction = multiply by reciprocal: 5 ÷ (3/2) = 10/3!

💡 Top tips

“Raise & divide” — add 1 to the power, divide by the new power.
Always rewrite first — roots become fractional powers, fractions become negative powers.
Don’t forget “+ c” — every indefinite integral needs one.
Check by differentiating your answer — you should get back the original function.
Expand brackets first for products. There’s no shortcut for ∫ uv dx at AA SL.
Watch ⁄ fractions: dividing by 3/2 = multiplying by 2/3.

⚠ Common mistakes

Forgetting “+ c” — automatic mark loss.
Dividing by the old power instead of the new one. ∫ x⁴ dx = x⁵/5, NOT x⁵/4.
Sign errors with negative powers. n = −3, n + 1 = −2, divide by −2 (not 2).
Trying to integrate products directly — always expand first.
Trying to use the rule for n = −1 — that needs ln, not the power rule.

You can now integrate any polynomial. Next: finding the constant of integration — when the question gives you a point and you can pin down c exactly.

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