Calculus/Further Methods of Integration/Contents

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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.

Review[edit]

Basic Integration Rules[edit]

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 Integration[edit]

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 Integrals[edit]

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)