Last modified on 3 December 2009, at 14:49

Calculus/Further Methods of Integration/Contents

TODO

Editor's note
The "Further Methods of Integration" section was orphaned in 2007. This is in the process of being merged into the main Calculus book.

ReviewEdit

Basic Integration RulesEdit

See Calculus/Definite integral.

\int 0\ du = C

\int ku\ du = k\times \int u\ du + C

\int (u + v)\ du = \int u\ du + \int v\ du + C

Partial IntegrationEdit

See Calculus/Integration techniques/Integration by Parts.

For two functions u and dv of a variable x,

\int u dv = u v - \int v du

where u is chosen by precedence according to LIPET:

  • Logarithmic
  • Inverse Trigonometric
  • Polynomial
  • Exponential
  • Trigonometric

Improper IntegralsEdit

See Calculus/Improper Integrals.

For any function f of variable x, continuous on the given infinite domain:

\int_{a}^{\infin} f(x)\, dx=\lim_{b \to \infin}\int_{a}^{b} f(x)\, dx

\int_{-\infin}^{b} f(x)\, dx=\lim_{a \to -\infin}\int_{a}^{b} f(x)\, dx

\int_{-\infin}^{\infin} f(x)\, dx=\int_{-\infin}^{c} f(x)\, dx + \int_{c}^{\infin} f(x)\, dx

For any function f of variable x continuous on the given interval, but with an infinite discontinuity at (1) a, (2) b, or some (3) c in [a,b]:

\int_{a}^{b} f(x)\, dx=\lim_{c \to b^-}\int_{a}^{c} f(x)\, dx (1)

\int_{a}^{b} f(x)\, dx=\lim_{c \to a^+}\int_{c}^{b} f(x)\, dx (2)

\int_{a}^{b} f(x)\, dx=\int_{a}^{c} f(x)\, dx+\int_{c}^{b} f(x)\, dx (3)