0

Hello could any one tell me some unusual or advanced integration techniques, I am already familiar with the standard ones like u-substitution, integration by parts, trig substitution, partial fractions, Feynman technique (differentiation under the integral), integrating the inverse, Laplace transforms In the integral and matrix inversion so I was wondering if anyone knew some rare ones (definite or indefinite)

Please do not answer with any already listed Otherwise Any help is appreciated Thanks in advance

Unmotivated L-function
  • 57
  • 6
  • Glasser's Master Theorem and Ramanujan's Master Theorem. –  Jul 08 '19 at 06:58
  • I found the technique for $\int_{-\infty}^{\infty}e^{-x^2}dx$ by squaring and using polar coordinates very elegant but I think it is specific to this particular integral. This also gives you an easy way to get the surface of the $n$ dimensional sphere. – quarague Jul 08 '19 at 07:05
  • Also related: https://math.stackexchange.com/questions/70974/lesser-known-integration-tricks – Hans Lundmark Dec 05 '19 at 08:24

1 Answers1

3

Two little known (perhaps?) methods for finding indefinite integrals are:

  1. The Rules of Bioche - Are rules used to guide one towards the most effective trigonometric substitution to use in integrals of the form $$\int f(\sin x, \cos x) \, dx,$$ where $f$ is a rational function of sine and cosine. For more details, see here or here.

  2. Ostrogradsky's Method - Is a method that finds the rational part of $$\int \frac{P(x)}{Q(x)} \, dx,$$ without having to find a factorisation for $Q$ and without having to decompose the integrand into partial fractions. Here $P$ and $Q$ are polynomials such that $\deg P < \deg Q$. For more details, see here.

omegadot
  • 12,364