# 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.

{\begin{aligned}&\int 0\,du=C\\&\int (k\cdot u)du=k\cdot \int u\,du+C\\&\int (u\pm v)du=\int u\,du\pm \int v\,du+C\end{aligned}}

### Partial Integration

See Calculus/Integration techniques/Integration by Parts.

For two functions $u$  and $dv$  of a variable $x$  ,

$\int u\,dv=uv-\int v\,du$

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:

{\begin{aligned}&\int \limits _{a}^{\infty }f(x)dx=\lim _{b\to \infty }\int \limits _{a}^{b}f(x)dx\\&\int \limits _{-\infty }^{b}f(x)dx=\lim _{a\to -\infty }\int \limits _{a}^{b}f(x)dx\\&\int \limits _{-\infty }^{\infty }f(x)dx=\int \limits _{-\infty }^{c}f(x)dx+\int \limits _{c}^{\infty }f(x)dx\end{aligned}}

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) $c\in [a,b]$  :

{\begin{aligned}\int \limits _{a}^{b}f(x)dx&=\lim _{c\to b^{-}}\int \limits _{a}^{c}f(x)dx&(1)\\\int \limits _{a}^{b}f(x)dx&=\lim _{c\to a^{+}}\int \limits _{c}^{b}f(x)dx&(2)\\\int \limits _{a}^{b}f(x)dx&=\int \limits _{a}^{c}f(x)dx+\int \limits _{c}^{b}f(x)dx&(3)\end{aligned}}