Subject: Calculus

List Of Integrals Of Rational Functions

Rational functions are any function which can be written as the ratio of two polynomial functions. Differentiation of rational functions has been easy because the calculation is straightforward. But that's not always the case for integration of rational function. The following are the list to the most common forms of rational integral. Enjoy!

  • \int (x+b)^r dx = \frac{[x + b]^{r+1}}{r+1} + C
  • \int (ax+b)^r dx = \frac{[ax + b]^{r+1}}{a(r+1)} + C
  • \int \frac{c}{x+b} dx = c\ln {|x + b|}+ C
  • \int \frac{(x+b)^r}{c} dx = \frac{1}{c}\cdot \frac{[x + b]^{r+1}}{r+1} + C
  • \int x(x+b)^r dx = \frac{ax(n+1)-b}{a^2(n+1)(n+2)}\cdot (ax+b)^{n+1} + C
  • \int \frac{x}{a+bx} dx = \frac{1}{b^2}[a + bx - a\ln{|a + bx|}] + C
  • \int \frac{x^2}{a+bx} dx = \frac{1}{b^3}\left [\frac{1}{2}(a + bx)^2 -2a(a+bx) + a^2\ln{|a + bx|}\right ] + C
  • \int \frac{x}{\left (a+bx\right )^2} dx = \frac{1}{b^2}\left [\frac{a}{a+bx} + \ln{|a + bx|}\right ] + C
  • \int \frac{x^2}{\left (a+bx\right )^2} dx = \frac{1}{b^3}\left [a + bx - \frac{a^2}{a+bx} - 2a\ln{|a + bx|}\right ] + C
  • \int \frac{x}{\left (a+bx\right )^3} dx = \frac{1}{b^2}\left [\frac{a}{2\left (a+bx\right )^2} - \frac{1}{a + bx}\right ] + C
  • \int \frac{dx}{x\left (a+bx\right )} = \frac{1}{a}\ln{\left |\frac{x}{a + bx}\right |} + C
  • \int \frac{dx}{x^2\left (a+bx\right )} = -\frac{1}{ax} +\frac{b}{a^2}\ln{\left |\frac{a+bx}{x}\right |} + C
  • \int \frac{dx}{x\left (a+bx\right )^2} = \frac{1}{a\left (a+bx\right )} + \frac{1}{a^2}\ln{\left |\frac{x}{a + bx}\right |} + C

Please click here for the list of irrational integrals.