Calculus/Further Methods of Integration/Contents

Review

Basic Integration Rules

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

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

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)