--- title: "Integration as the Reverse of Differentiation" description: "Recover a function from its derivative by raising the power and dividing, and carry the constant of integration that differentiation destroyed. Use a known point to pin that constant down." canonical: https://lightmysky.com/learn/mathematics/integration-as-the-reverse-of-differentiation-mt_edtk3ArxRk source: https://lightmysky.com/learn/mathematics/integration-as-the-reverse-of-differentiation-mt_edtk3ArxRk.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Integration as the Reverse of Differentiation Recover a function from its derivative by raising the power and dividing, and carry the constant of integration that differentiation destroyed. Use a known point to pin that constant down. Subject: Mathematics · Area: Calculus & Analysis · Ages 17 to 18 Page: https://lightmysky.com/learn/mathematics/integration-as-the-reverse-of-differentiation-mt_edtk3ArxRk ## Ready when they can - Integrate 6x² - 4x and check the answer by differentiating it - Explain why general indefinite integrals include + c, why antiderivatives on one interval differ by a constant, and why c cancels in F(b) - F(a) - Find the particular function whose derivative is 2x and which passes through (1, 5) ## Lesson: Integration as the Reverse of Differentiation You have used the power rule to find a derivative. Integration asks for a function with that derivative. For the Ridgeway tank, let V be volume and t be time. If water leaves at rate R with no inflow, dV/dt = -R. Recovering V requires integration and a known volume. Check an antiderivative by differentiating it; you can also evaluate that derivative to check a gradient. For n ≠ -1, ∫xⁿ dx = xⁿ⁺¹/(n + 1) + c: raise the power and divide by the new power. Divide coefficients too; ∫5x³ dx = 5x⁴/4 + c. Integrate sums term by term, and ∫k dx = kx + c. Negative and fractional powers follow this rule where they are defined. For example, ∫x^(-3) dx = -x^(-2)/2 + c on either side of zero. Expand products first; multiplying antiderivatives does not integrate a product. *(drawing: The new power becomes the divisor. The constant names the whole family.)* **Example.** Integrate 6x² - 4x. The first term gives 6x³/3 = 2x³. The second gives -4x²/2 = -2x². Thus ∫(6x² - 4x) dx = 2x³ - 2x² + c. Differentiate to check: the two terms return 6x² and -4x, while c gives zero. For a fractional power, ∫x^(1/3) dx = (3/4)x^(4/3) + c because the new power is 4/3. An indefinite integral names all antiderivatives on one interval. They differ only by a constant: if F' = G', then (F - G)' = 0, so F - G is constant there. For 2x, the family is x² + c. Differences such as F(b) - F(a) cancel c. An excluded point can split the domain into separate intervals, each with its own constant. For x⁻², the intervals x < 0 and x > 0 need not use the same c. *(drawing: Each expression differentiates to 2x.)* **Example.** Suppose dy/dx = 2x and the curve passes through (3, 11). Integrate first: y = x² + c. Substitute the coordinates: 11 = 9 + c, so c = 2 and y = x² + 2. The point fixes the constant on this interval. To find another value, substitute its x-coordinate into the completed formula. At x = 4, y = 16 + 2 = 18. **Recap.** For n ≠ -1, raise the power and divide by the new power. Add c for the antiderivative family on an interval. A known point fixes c. Differentiate to check. *Lesson and practice text written for this dataset.* ## Practice 34 questions on this page, each with its working shown. ## Needs first - [The Power Rule for Differentiating Polynomials](https://lightmysky.com/learn/mathematics/the-power-rule-for-differentiating-polynomials-mt_l7f4j44bR3) ## Opens up - [Kinematics with Calculus](https://lightmysky.com/learn/mathematics/kinematics-with-calculus-mt_2kNspOOoDw) - [Recovering Motion by Integrating Acceleration](https://lightmysky.com/learn/science/recovering-motion-by-integrating-acceleration-mt_CXo7wKOA46) - [The Fundamental Theorem of Calculus](https://lightmysky.com/learn/mathematics/the-fundamental-theorem-of-calculus-mt_EsS_ovdz2A) - [The Centre of Mass and the Motion of a System](https://lightmysky.com/learn/science/the-centre-of-mass-and-the-motion-of-a-system-mt_fHO_KrMNKo) - [The Definite Integral and the Area Under a Curve](https://lightmysky.com/learn/mathematics/the-definite-integral-and-the-area-under-a-curve-mt_S2IO2S7kgd) - [What a Differential Equation Says and What a Solution Is](https://lightmysky.com/learn/mathematics/what-a-differential-equation-says-and-what-a-solution-is-mt_tHlHg2nK3o) - [Separable Equations](https://lightmysky.com/learn/mathematics/separable-equations-mt_itSTsLTkwQ) - [Integration by Parts](https://lightmysky.com/learn/mathematics/integration-by-parts-mt_v6jZVs2_tf)