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§ Calculus

Integration

§ Calculus

Integration

CCSS.HSF.IF3 min read

Integration is the mathematical process of finding the antiderivative of a function, essentially reversing differentiation. The power rule for integration states that ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, where C represents the constant of integration. Definite integrals evaluate between specific bounds and represent the area under a curve.

§ 01

Why it matters

Integration appears throughout science and engineering to calculate accumulated quantities. In physics, integrating velocity over time gives displacement — if a car travels at 60 mph for 3 hours, integration shows it moved 180 miles total. Engineers use integration to find volumes of complex shapes, like calculating that a cone with radius 4 and height 6 has volume 32π cubic units. Economics relies on integration to determine total profit from marginal profit functions. Medical imaging uses integration algorithms to reconstruct CT scans from thousands of X-ray measurements. In advanced mathematics, integration connects to differential equations, Fourier analysis, and probability theory, making it essential for students progressing to calculus-based courses in STEM fields.

§ 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: ∫ x dx

Answer: x2/2 + C

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

Find the integral: ∫ (2 x2 - x + 2) dx

Answer: 2 x3/3 - x2/2 + 2 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: ∫ 2 x2 dx → 2 x3/3 — Exponent 2 becomes 3, divide by 3: 2x³/3 = 2 x^3/3.
  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: ∫ 2 dx → 2 x — The integral of a constant k is kx.
  5. Combine and add C → 2 x3/3 - x2/2 + 2 x + C — Add all terms together. Always include the integration constant C.
Medium§ 03

Find the integral: ∫ sin(x) dx

Answer: - cos(x) + C

  1. Apply the rule: ∫sin(x) dx = −cos(x) → - cos(x) + C — The constant 1 is carried through the integration.
§ 04

Common mistakes

  • Forgetting the constant of integration C in indefinite integrals, writing ∫x² dx = x³/3 instead of x³/3 + C
  • Incorrectly applying the power rule to n = -1, writing ∫x⁻¹ dx = x⁰/0 instead of recognizing this equals ln|x| + C
  • Evaluating definite integrals backwards, computing ∫₁³ x dx as 1²/2 - 3²/2 = -4 instead of 3²/2 - 1²/2 = 4
  • Mishandling negative exponents in the power rule, writing ∫x⁻² dx = x⁻¹/(-1) instead of -x⁻¹ + C = -1/x + C
§ 05

Frequently asked questions

What is the difference between definite and indefinite integrals?
Indefinite integrals have no bounds and include a constant C, representing a family of functions. Definite integrals have upper and lower bounds, produce a specific numerical value, and represent the area under a curve between those points.
Why do we add the constant C in integration?
The constant C accounts for the fact that many functions have the same derivative. Since d/dx(x² + 5) = d/dx(x² + 10) = 2x, the antiderivative of 2x could be x² plus any constant value.
How do you check if an integration answer is correct?
Differentiate the result and verify it matches the original integrand. For example, if ∫2x dx = x² + C, then d/dx(x² + C) = 2x, confirming the integration is correct.
What does it mean when the power rule doesn't apply?
The power rule fails when n = -1 because dividing by zero is undefined. Instead, ∫x⁻¹ dx = ∫(1/x) dx = ln|x| + C, which is a special case requiring the natural logarithm function.
How do definite integrals relate to area under curves?
The definite integral ∫ₐᵇ f(x) dx equals the signed area between the curve f(x) and the x-axis from x = a to x = b. Areas above the x-axis are positive, while areas below are negative.
§ 06

See also

FunctionAreaRadiusVolumeProbabilityMeanLogarithm
§ 06

Where to next?

Prerequisites
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