Overview

Integration is the reverse process of differentiation. Antiderivatives allow you to recover an original function from its derivative, and are key to calculating area under curves, motion from acceleration, and accumulated change.

Key Concepts and Structures

• Antiderivative: A function F(x) such that F'(x) = f(x)
• Indefinite Integral: \int f(x) dx = F(x) + C, where C is the constant of integration
• Basic Integration Rules:
  • \int x^n dx = \frac{x^{n+1}}{n+1} + C (for n ≠ -1)
  • \int \frac{1}{x} dx = \ln|x| + C
  • \int e^x dx = e^x + C; \int a^x dx = \frac{a^x}{\ln a} + C
  • \int \sin x dx = -\cos x + C; \int \cos x dx = \sin x + C
• u-Substitution: Use when an integral matches a chain rule pattern. Let u = g(x) so du = g'(x) dx.
• Common Applications: Recover velocity from acceleration, find position from velocity, and calculate accumulated quantities.

Quick Tip

Practice Problems

1. Find the general antiderivative of f(x) = 6x^2
   Show Solution

   \int 6x^2 dx = 6 \cdot \int x^2 dx = 6 \cdot \frac{x^3}{3} + C = 2x^3 + C

   Final Answer: 2x^3 + C

2. Evaluate \int (2x)(x^2 + 1)^5 dx using substitution
   Show Solution

   Let u = x^2 + 1 → du = 2x dx

   So the integral becomes \int u^5 du = \frac{u^6}{6} + C

   Substitute back: \frac{(x^2 + 1)^6}{6} + C

   Final Answer: \frac{(x^2 + 1)^6}{6} + C