# 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 {\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 ${\displaystyle u}$ and ${\displaystyle dv}$ of a variable ${\displaystyle x}$ ,

${\displaystyle \int u\,dv=uv-\int v\,du}$

where ${\displaystyle u}$ is chosen by precedence according to LIPET:

• Logarithmic
• Inverse Trigonometric
• Polynomial
• Exponential
• Trigonometric

### Improper Integrals

See Calculus/Improper Integrals.

For any function ${\displaystyle f}$ of variable ${\displaystyle x}$ , continuous on the given infinite domain:

{\displaystyle {\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 ${\displaystyle f}$ of variable ${\displaystyle x}$ continuous on the given interval, but with an infinite discontinuity at (1) ${\displaystyle a}$ , (2) ${\displaystyle b}$ , or some (3) ${\displaystyle c\in [a,b]}$ :

{\displaystyle {\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}}}