Calculus/Further Methods of Integration/Contents

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) $c\in [a,b]$ :

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