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Indefinite Integrals

8.1 Anti-Differentiation Concepts

Anti-Differentiation

  • The area under a graph within an interval x \in [a,b] can be estimated by a sum of rectangles for smaller intervals of v.
  • Differentiation is the process of finding the derivative function from the original function. Conversely, the process of finding a function from its derivative is called anti-differentiation.
  • f(x) is known as the antiderivative of f^{\prime}(x).
  • A derivative function might have several antiderivatives (see example).
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6.1 Antiderivatives of Functions

Antiderivatives of Specific Functions

  • We will be looking at antiderivatives of polynomial, exponential and trigonometric functions.
  • The formulas are as below:
i) Polynomial functions

\int x^{n} d x=\frac{1}{n+1} x^{n+1}+c \quad,\ n \neq-1

\int x^{-1} d x=\int \frac{1}{x} d x=\ln |x|+c

Note: Notice that in there is an absolute value over to make the result fit the domain for all logarithmic functions is (0,\ +\infty), and x=0 is not possible in the first place as it would make \frac{1}{x} undefined.

ii) Exponential functions
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