1: Review of antiderivatives

Section 1: Integration Techniques - 1: Review of Antiderivatives

A function $F(x)$ is an antiderivative of $f(x)$ if $F'(x) = f(x)$. Finding antiderivatives, or indefinite integration, is the reverse process of differentiation, symbolized by the integral sign: $\int f(x) dx = F(x) + C$. The constant of integration $C$ is crucial, representing the infinite family of functions differing by a constant that all share the same derivative.

Mastering basic antiderivative formulas is essential. Recall these fundamental results:

• Power Rule (for $n \neq -1$): $\int x^n dx = \frac{x^{n+1}}{n+1} + C$
• Exponential Functions:
  $\int e^x dx = e^x + C$,
  $\int a^x dx = \frac{a^x}{\ln a} + C$ ($a > 0$, $a \neq 1$)
• Reciprocal: $\int \frac{1}{x} dx = \ln|x| + C$
• Trigonometric Functions:
  • $\int \sin(x) dx = -\cos(x) + C$
  • $\int \cos(x) dx = \sin(x) + C$
  • $\int \sec^2(x) dx = \tan(x) + C$
  • $\int \csc^2(x) dx = -\cot(x) + C$
  • $\int \sec(x)\tan(x) dx = \sec(x) + C$
  • $\int \csc(x)\cot(x) dx = -\csc(x) + C$
• Hyperbolic Functions:
  $\int \cosh(x) dx = \sinh(x) + C$,
  $\int \sinh(x) dx = \cosh(x) + C$
• Constants: $\int k dx = kx + C$ ($k$ constant)

Integration is linear. This means you can integrate term-by-term and pull constants outside the integral:
$\int [c \cdot f(x) \pm g(x)] dx = c \cdot \int f(x) dx \pm \int g(x) dx$ (where $c$ is a constant).

The Fundamental Theorem of Calculus, Part 1 links antiderivatives directly to definite integrals. If $F(x)$ is an antiderivative of $f(x)$ (i.e., $F'(x) = f(x)$) and $f(x)$ is continuous on $[a, b]$, then:
$\int_{a}^{b} f(x) dx = F(b) - F(a)$
This theorem transforms the problem of evaluating the net area under a curve into finding an antiderivative and computing a difference. Always verify the function is continuous over the interval.