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§ Calculus

Integration

§ Calculus

Integration

CCSS.HSF.IF3 min read

Integration is the mathematical operation that reverses differentiation, finding a function whose derivative equals the given expression. The power rule for integration states that ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, where C represents the constant of integration. Definite integrals, evaluated between specific bounds, calculate the exact area under a curve.

§ 01

Why it matters

Integration calculates accumulated quantities from rates of change, making it essential across science and engineering. Physicists use integration to find displacement from velocity data — if a car travels at 20 m/s for 5 seconds, integration gives the distance of 100 metres. Engineers apply definite integrals to determine water flow through pipes, calculating that a flow rate of 3 m³/min over 4 minutes delivers 12 cubic metres. In economics, integrating marginal cost functions reveals total production costs. Year 12 students encounter definite integrals for area calculations, whilst Year 13 explores advanced techniques like integration by parts and substitution. These methods solve separable differential equations, modelling population growth, radioactive decay, and compound interest in real-world scenarios.

§ 02

How to solve integration

Integration

  • Integration is the reverse of differentiation.
  • Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1).
  • Definite integral: evaluate at upper and lower bounds, subtract.
  • The definite integral gives the area under the curve.

Example: ∫x² dx = x³/3 + C. ∫₁² x² dx = 83 − 13 = 73.

§ 03

Worked examples

Beginner§ 01

Find the integral: ∫ 2 x2 dx

Answer: 2 x3/3 + C

  1. Apply the power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) → ∫ 2 x2 dx = 2·x3/3 — Increase the exponent by 1 (to 3) and divide by the new exponent.
  2. Simplify and add constant → 2 x3/3 + C — Always add the constant of integration C for indefinite integrals.
Easy§ 02

Find the integral: ∫ (3 x2 + x + 5) dx

Answer: x3 + x2/2 + 5 x + C

  1. Write out the rule → ∫xⁿ dx = xⁿ⁺¹/(n+1) — The power rule for integration: raise the exponent by 1 and divide by the new exponent.
  2. Integrate the first term: ∫ 3 x2 dx → x3 — Exponent 2 becomes 3, divide by 3: 3x³/3 = x^3.
  3. Integrate the second term: ∫ x dx → x2/2 — Exponent 1 becomes 2, divide by 2: 1x²/2 = x^2/2.
  4. Integrate the constant: ∫ 5 dx → 5 x — The integral of a constant k is kx.
  5. Combine and add C → x3 + x2/2 + 5 x + C — Add all terms together. Always include the integration constant C.
Medium§ 03

Find the integral: ∫ ex dx

Answer: ex + C

  1. Apply the rule: ∫ex dx = ex → ex + C — The constant 1 is carried through the integration.
§ 04

Common mistakes

  • A common error is forgetting the constant of integration, writing ∫x² dx = x³/3 instead of x³/3 + C for indefinite integrals.
  • Another mistake involves incorrect power rule application, calculating ∫x³ dx = x³/3 instead of x⁴/4 by failing to increase the exponent first.
  • Students often mishandle definite integrals, computing ∫₁³ x² dx = 9/3 - 1/3 = 8/3 instead of correctly evaluating [x³/3]₁³ = 27/3 - 1/3 = 26/3.
§ 05

Frequently asked questions

What is the difference between definite and indefinite integrals?
Indefinite integrals include a constant of integration C and represent a family of functions, whilst definite integrals have numerical bounds and calculate specific values. For example, ∫x dx = x²/2 + C is indefinite, but ∫₀² x dx = 2 is definite.
How do you check if an integration answer is correct?
Differentiate the result to verify it equals the original function. If ∫2x dx = x² + C, then d/dx(x² + C) = 2x confirms the answer. For definite integrals, check the arithmetic when substituting bounds.
Why does the power rule not work for x⁻¹?
When n = -1, the formula xⁿ⁺¹/(n+1) gives x⁰/0, which is undefined. Instead, ∫x⁻¹ dx = ∫1/x dx = ln|x| + C. This logarithmic function has the special property that its derivative equals 1/x.
What does the area under a curve represent physically?
The physical meaning depends on the axes. If y represents velocity and x represents time, the area gives displacement. For force versus distance, area equals work done. In general, area represents the accumulation of the y-quantity over the x-interval.
How do you integrate trigonometric functions?
Basic trigonometric integrals follow standard patterns: ∫sin x dx = -cos x + C and ∫cos x dx = sin x + C. More complex trigonometric functions require substitution methods or integration by parts, typically covered in Year 13 mathematics courses.
§ 06

See also

FunctionExpressionArea
§ 06

Where to next?

Prerequisites
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