Indefinite Integrals
Antiderivatives
\(\int f(x)dx = F(x)+C\) where \(F'(x)=f(x)\).
Integration Rules
| Property | Statement |
|---|---|
| Power | \(\int x^n dx=\frac{{x^{{n+1}}}}{{n+1}}+C\) (n≠−1) |
| Exponential | \(\int e^x dx=e^x+C\) |
| Trig | \(\int\sin x\,dx=-\cos x+C\) |
Examples
Example 1. Find \(\int(3x^2+2x)dx\).
Solution. \(x^3+x^2+C\).
Deep Dive: Indefinite Integral
This section builds durable understanding of indefinite integral in calculus through definition-first reasoning, theorem mapping, and error-checking workflows.
Use a two-pass method: first derive the structure symbolically, then validate with a concrete numerical or geometric test case.
Visual Intuition
Convert algebra into a diagram, graph, or dependency map before solving. Visual-first analysis reduces sign errors and makes assumptions explicit.
Practice Set
Practice A. Re-derive one key formula on this page from first principles and annotate each transformation.
Target. Your final line should include assumptions, derivation path, and a quick verification.
Practice B. Build an application scenario using indefinite integral and solve it with both symbolic and numeric methods.
Target. Compare outputs and explain any approximation gap.
References & Editorial Notes
- Stewart, Calculus.
- Strang, Introduction to Linear Algebra.
- Apostol, Mathematical Analysis.
Editorial update: Reviewed on 2026-04-14 for notation consistency, conceptual clarity, and exercise quality.