Calculus/Further Methods of Integration/Contents

Basic Integration Rules

 * See Calculus/Definite integral.

$$\begin{align} &\int0\,du=C\\ &\int(k\cdot u)du=k\cdot\int u\,du+C\\ &\int(u\pm v)du=\int u\,du\pm\int v\,du+C \end{align}$$

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:

$$\begin{align} &\int\limits_a^\infty f(x)dx=\lim_{b\to\infty}\int\limits_a^b f(x)dx\\ &\int\limits_{-\infty}^b f(x)dx=\lim_{a\to-\infty}\int\limits_a^b f(x)dx\\ &\int\limits_{-\infty}^\infty f(x)dx=\int\limits_{-\infty}^c f(x)dx+\int\limits_c^\infty f(x)dx \end{align}$$

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) $$c\in[a,b]$$ :

$$\begin{align} \int\limits_a^b f(x)dx&=\lim_{c\to b^-}\int\limits_a^c f(x)dx&(1)\\ \int\limits_a^b f(x)dx&=\lim_{c\to a^+}\int\limits_c^b f(x)dx&(2)\\ \int\limits_a^b f(x)dx&=\int\limits_a^c f(x)dx+\int\limits_c^b f(x)dx&(3) \end{align}$$