# Calculus/Further Methods of Integration/Contents

 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

### Basic Integration Rules

See Calculus/Definite integral.

${\displaystyle \int 0\ du=C}$

${\displaystyle \int ku\ du=k\times \int u\ du+C}$

${\displaystyle \int (u+v)\ du=\int u\ du+\int v\ du+C}$

### Partial Integration

See Calculus/Integration techniques/Integration by Parts.

For two functions u and dv of a variable x,

${\displaystyle \int udv=uv-\int vdu}$

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:

${\displaystyle \int _{a}^{\infty }f(x)\,dx}$=${\displaystyle \lim _{b\to \infty }\int _{a}^{b}f(x)\,dx}$

${\displaystyle \int _{-\infty }^{b}f(x)\,dx}$=${\displaystyle \lim _{a\to -\infty }\int _{a}^{b}f(x)\,dx}$

${\displaystyle \int _{-\infty }^{\infty }f(x)\,dx}$=${\displaystyle \int _{-\infty }^{c}f(x)\,dx+\int _{c}^{\infty }f(x)\,dx}$

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]:

${\displaystyle \int _{a}^{b}f(x)\,dx}$=${\displaystyle \lim _{c\to b^{-}}\int _{a}^{c}f(x)\,dx}$ (1)

${\displaystyle \int _{a}^{b}f(x)\,dx}$=${\displaystyle \lim _{c\to a^{+}}\int _{c}^{b}f(x)\,dx}$ (2)

${\displaystyle \int _{a}^{b}f(x)\,dx}$=${\displaystyle \int _{a}^{c}f(x)\,dx+\int _{c}^{b}f(x)\,dx}$ (3)