Integral Calculus

consider continuity, smoothness, etc.

Definition of an improper integral \int(a to b) f(x) dx is an improper integral if

  1. f becomes infinite at one or more points of the interval of integration, or
  2. one or both of the limits of integration is infinite, or
  3. both 1 and 2 hold

Summation

Types of infinity

Constant integral \(\int 1dx = x + C\)

Area under the curve \(\int a dx = ax + C\) \(\int x^n dx = \frac{x^{n+1}}{n+1} + C, n \neq -1\)

Integrating trigonometry \(\int sin(x) dx = -cos(x) + C\) \(\int cos(x) dx = sin(x) + C\) \(\int sec^2(x) dx = tan(x) + C\) \(\int csc^2(x) dx = -cot(x) + C\) \(\int sec(x)(tan(x)) dx = sec(x) + C\) \(\int csc(x)(cot(x)) dx = -csc(x) + C\)

U-substitution

Integration by parts

Indefinite integral

Definite integral

Partial fraction

Taylor series Taylor series gives us a way to approximate around regions of interest.

Series

Converges

Diverges

Radius of convergence, a power series will converge for certain value of x.

  • Arithmetic series
  • Geometric series
  • Power series
  • Taylor series
  • Mclaurin series
  • Harmonic series

Calculus 2 Problems and Solutions

\[\int_{-1}^{1} \frac{1}{\sqrt[3](x)} dx\]

The trick is to rewrite the cube root, then integrate from -1 to 0 and 0 to 1

The first integral should go from -1 to b, as limit of b goes to \(0^-\) and the second integral should go from c to 1, as limit of c goes to \(0^+\)