Integrals of Rational Functions

Key Questions

How do I find the integral of a rational function?

Answer:

There is no one method that works for all rational functions.

Explanation:

Among the methods we can try are:

Substitution, Parts, Partial Fractions, Trigonometric Substitution, Miscellaneous Substitution.

Jim H · 1 · Mar 9 2018
How do I divide one polynomial by another?

Use long division!

I hope that this was helpful.

Wataru · · Oct 19 2014
What is a rational function?

Definition

A rational function $r(x)$ is any function of the form

$r(x)={p(x)}/{q(x)}$ ,

where $p(x)$ and $q(x)$ are polynomial functions.

I hope that this was helpful.

Wataru · · Oct 16 2014

Questions

How do you integrate $(x+1)/(x^2+2x+1)$ ?
How do you integrate $x/(1+x^4)$ ?
How do you integrate $dx / (2sqrt(x) + 2x$ ?
What is the integration of $1/x$ ?
How do you integrate $(1+x)/(1-x)$ ?
How do you integrate $(2x^3-3x^2+x+1)/(-2x+1)$ ?
How do you find integral of $((secxtanx)/(secx-1))dx$ ?
How do you integrate $(6x^5 -2x^4 + 3x^3 + x^2 - x-2)/x^3$ ?
How do you integrate $((4x^2-1)^2)/x^3dx$ ?
How do you integrate $(x+3) / sqrt(x) dx$ ?
How do you find the integral of $(x^4+x-4) / (x^2+2)$ ?
How do you integrate $int (x^4)/(x^(5)+1)$ ?
How do you evaluate: indefinite integral $(1+x)/(1+x^2) dx$ ?
How do you integrate $((x^2) / (16-x^3)^2) dx$ ?
How do you integrate $(x^3+2x^2-x) / x dx$ ?
How do you integrate $1/ (x^2 -2x + 17)$ ?
How do you integrate $(5)/(1+3x)^3$ ?
What is the integral of 1/(2x)?
What is the integral of $1/(x^5)$ ?
What is the integral from 0 to 16 of $1/(16-x) dx$ ?
What is the integral of $x/(x-2)$ from 0 to 3?
What is the integral from 0 to1 of $(4/(t^2+1))dt$ ?
How do you find the integral of $(x^3-4x-10)/(x^2-x-6)$ from 0 to 1?
How do you integrate $[(x + 4)/(x^2 + 8 x +17)] dx$ ?
How do you integrate $(1+x) /(1+ x)^2 dx$ ?
How do you find the integral of $(4x)/(4x+7)dx$ ?
What is the Integral of $(x+1)/x dx$ ?
How do you integrate $(xdx)/(x^2+2)^3$ which has the upper and lower limits, 2 and 1 ?
How do you integrate $36/(2x+1)^3$ ?
How do you integrate $(8x)/(4x^2+1)$ ?
How do you integrate $(x+2)/(x+3)$ ?
How do you find the antiderivative of $(x^(3) + 2x)/(x^(4) + 4x^(2) +2)$ ?
How do you find the antiderivative of $(x^5-x^3+2x)/x^4$ ?
How do you find the antiderivative $f(x)=6/x^5$ ?
How do you find the antiderivative of $4/x^2$ ?
What is the antiderivative of $(1/(2x))$ ?
How do you find the antiderivative of $1/(1-x)^2$ ?
How do you find the antiderivative of $1/x^2$ ?
How do you find the anti-derivative of $3 / x^3$ ?
How do you find the antiderivative of $-7/(x^2+36)$ ?
How do you calculate the antiderivative of $(-2x^3+14x^9)/(x^-2)$ ?
What is the antiderivative of $(5/x)^2$ ?
How do I find the antiderivative of $x / (x^2 +1)^2$ ?
How do I find the antiderivative for $(1/sqrt(x) + 2/x^3) dx$ ?
What is the antiderivative of $(2+x^2)/(1+x^2)$ ?
What is the antiderivative of $1/(1+x^2)$ ?
What is the antiderivative of $1/ x^2$ ?
What is the antiderivative of $(5x)/(x^2+1)$ ?
Consider the function $f(x)=(10/x^2)-(7/x^6)$ if f(1)=0, then what is f(x)?
What is the antiderivative of $(1)/(1+x^2)$ ?
What is the antiderivative of $x/(x+1)$ ?
What is the antiderivative of $(2+x^2)/(1+x^2)$ ?
How do you find the integral of $1/(2x) dx$ ?
How do you find the integral of $1/(x^2+4)$ ?
How do you find the integral of $x^3/(1+x^2)$ ?
How do you find the integral of $dx/(1+x^2)$ ?
How do you find the integral of $(6x+1)/(x^2+2x+3)$ ?
How do you integrate $int (1+x) /(1+ x)^2 dx$ ?
Question #e30bc
What is $int_3^oo 3/x -2/(x-2)dx$ ?
What is $int_-oo^oo (x^2)(e^(-x^3))dx$ ?
What is $int_-oo^oo e^(-x^2)dx$ ?
$int (4x)/(1-x^4)dx$ =?
What is the antiderivative of $7/x$ ?
What is the antiderivative of $x/(x+1)$ ?
What is $int (4-x ) / (x^3-6x +4 )$ ?
What is $int (x^2-5x+4 ) / (x^3-2x +4 )$ ?
What is $int (x^2-5x+4 ) / (x^3-2x +1 )$ ?
What is $int (-3x^2-5x+4 ) / (x^3-2x +1 )$ ?
What is $int (-3x^2+4 ) / (5x^2-2x +1 )$ ?
What is $int (-3x^3+4 ) / (5x^2-2x +1 )$ ?
What is $int (-3x^3+4 ) / (-2x^2-2x +1 )$ ?
What is $int (-3x^3+4 ) / (-2x^2+x +1 )$ ?
What is $int (-3x^3+4 ) / (-2x^2+x +7 )$ ?
What is $int (-2x^3+4 ) / (-2x^2+x +7 )$ ?
What is $int (-2x^3+4x ) / (-2x^2+x +7 )$ ?
What is $int (-2x^3-x ) / (-2x^2+x +7 )$ ?
What is $int (-2x^3-x ) / (-4x^2+2x +7 )$ ?
What is $int (-2x^3-x)/(-4x^2+2x+3) dx$ ?
What is $int (-2x^3-x^2-5x+2 ) / (-4x^2+2x +3 )$ ?
What is $int (-2x^3-x^2+x+2 ) / (3x^2+2x +3 )$ ?
What is $int (-2x^3-x^2+x+2 ) / (2x^2- x +3 )$ ?
What is $int (-2x^3-x^2+6x+9 ) / (2x^2- x +3 )$ ?
What is $int (-2x^3-2x^2+6x+9 ) / (2x^2- x +3 )$ ?
What is $int (x^3-2x^2+6x+9 ) / (2x^2- x +3 )$ ?
What is $int (x^3-2x^2+6x+9 ) / (-x^2- x +3 )$ ?
What is $int (x^3-2x^2+6x+9 ) / (-x^2- x +2 )$ ?
What is $int (x^3-2x^2+6x+9 ) / (-x^2+ 9 x +2 )$ ?
What is $int (x^3-2x^2+6x-3 ) / (-x^2+ 9 x +2 )$ ?
What is $int (x^3-2x^2+5x-3 ) / (-x^2+ 9 x +2 )$ ?
What is $int (x^3-2x^2+5x-3 ) / (-x^2+ 7 x +2 )$ ?
What is $int (x^3-2x^2+5x-3 ) / (-x^2+ 3 x -4 )dx$ ?
What is $int (x^3-3x^2+5x-3 ) / (-6x^2+ 3 x -4 )$ ?
What is $int (2x^3-3x^2-4x-3 ) / (-6x^2+ 3 x -4 )dx$ ?
What is $int (2x^3-3x^2-4x-3 ) / (-5x^2+ 2 x -4 )dx$ ?
What is $int (2x^3-3x^2-4x-3 ) / (-8x^2+ 2 x -2 )$ ?
What is $int (2x^3-3x^2-2x-3 ) / (-8x^2+ 2 x -2 )$ ?
What is $int (2x^3-2x-3 ) / (-x^2+ 5 x -2 )$ ?
What is $int (2x^3-2x-3 ) / (-x^2+ 5 x )$ ?
What is $int (2x^3-2x-3 ) / (-x^4+ 5 x )$ ?
What is $int (-x^3-2x-3 ) / (7x^4+ 5 x -1 )$ ?
What is $int (-x^3+4x^2-4 ) / (-8x^4+ 2 x -1 )$ ?
What is $int (-x^3-x-1 ) / (-2x^2+ 3 x +5 )$ ?
What is $int (-x^3+9x-1 ) / (-2x^2+ 3 x +5 )$ ?
What is $int (-x^3+9x-1 ) / (-2x^2- 3 x +5 )$ ?
What is the antiderivative of $2/x$ ?
What is the antiderivative of $2/(x^2+1)$ ?
What is the antiderivative of $x/(1-x)^4$ ?
What is the antiderivative of $x/(7x^2+3)^5$ ?
What is the antiderivative of $1/(2x)$ ?
What is the antiderivative of $(5/x)^2$ ?
What is the antiderivative of $(x+2)/(x+1)$ ?
What is the integral of $(x^2 +2x-1)/(x^2+9)$ ?
What is the integral of $int (2x-5)/(x^2+2x+2)$ ?
What is the integral of $int 1/(x^(3/2) + x^(1/2)) dx$ ?
What is the integral of $int (3x+1) / (2x^2 -6x +5))dx$ ?
What is the integral of $int 1 / (sqrt(x+1) - sqrt(x))$ ?
What is the integral of $int ( 1 / (25 + x^2) ) dx$ ?
What is the integral of $int (x)/(x+1)^3 dx$ ?
What is the antiderivative of $(ln(x))^2$ ?
Question #a407d
How do you integrate $(2x)/(x²-1)$ ?
How do you integrate $(1/x)$ ?
How do you integrate $x^3/(1+x^2)$ ?
How do you integrate $x^2/(x^2+1)$ ?
How do you integrate $x/(x+1)dx$ ?
How do you integrate $1/x^2$ ?
How do you integrate $1/(x^2+4)$ ?
How do you integrate $x/(x+1)^3$ ?
How do you integrate $(x^3+x^2+2x+1)/((x^2+1)(x^2+2)) dx$ ?
How do you integrate $x/(x+10)$ ?
How do you integrate $(x^2+2x-1)/(x^3-x)dx$ ?
How do you integrate $(x^(1/3))/(((x^(1/3))-1))$ ?
How do you integrate $x/(x-6)$ ?
How do you integrate $(x-2)/(x-1)$ ?
How do you integrate $1 / (x^5)$ ?
How do you integrate $(x)/(x+10) dx$ ?
How do you integrate $x^3 / (4-x^2)$ ?
Question #8a7b6
How do you integrate $1/(x^2+4)$ ?
How do you integrate $1/(x^2+x) dx$ ?
How do you integrate $(x^2 + 3x +1)/(x^2 - x - 6)$ ?
How do you integrate $(x^3)/(1+x^2)$ ?
How do you integrate $x/(x^2+1)$ ?
How do you integrate $(x^3)(x^2+1)^(1/2)$ ?
How do you integrate $x^2/((x-3)(x+2)^2)$ ?
How do you integrate $1/(x^2 - 3)$ ?
How do you integrate $(x^2 + 2x -5) / [(x - 2)*(x+1)*(x^2+1)]$ ?
How do you integrate $(1/(16+x^2))$ ?
How do you integrate $2/x$ ?
How do you integrate $(x/(x+2))$ ?
How do you integrate $1/(x^2 + 9)$ ?
How do you integrate $(x + 2)^2 / x^(1/3)$ ?
How do you integrate $(6x+1)/(x^2+2x+3)$ ?
How do you integrate $[(x^2 + 1)/(x^2 - 1)]dx$ ?
How do you integrate $1/((1+x^2)^2)$ ?
How do you integrate $(x-2)/(x-1)$ ?
How do you integrate $[1/(x^2-4)] dx$ ?
How do you integrate $(-2x)/((x^2+2x+2)^2)$ ?
How do you integrate $(x^2+2x-1) / (x^3 - x)$ ?
What is the integral of $x^3/(x^2+1)$ ?
How do you integrate $((x^3 + 7) / x^2) dx$ ?
How do you integrate $(5/sqrt x) dx$ ?
How do you integrate $(1/x^4) dx$ ?
What is the antiderivative of $1 / (x^2)$ ?
What is the antiderivative of $4/x^2$ ?
What is the antiderivative of $1/(x^2+4)$ ?
What is the antiderivative of $x^3/(1+x^2)$ ?
What is the antiderivative of $x sqrtx$ ?
What is the antiderivative of $1/ x^2$ ?
What is the antiderivative of $x^3/(x^4-5)^3$ ?
What is the antiderivative of $x/(1+x^4)$ ?
What is the antiderivative of $(x^2-1) / (x^2+1)$ ?
What is the antiderivative of $3/(x^2+1)$ ?
What is the antiderivative of $(5x)/(x^2+1)$ ?
What is the antiderivative of $(e^x)/x$ ?
What is the antiderivative of $x/(x^2+1)$ ?
What is the antiderivative of $1/(x^2-x)$ ?
What is the antiderivative of $x^2/2$ ?
What is the antiderivative of $(5/x)^2$ ?
What is the antiderivative of $1 / (x^5)$ ?
What is the antiderivative of $9/x^3$ ?
What is the antiderivative of $1/x^3$ ?
Can anyone explain the concept of derivatives?
How do you integrate $4/((x+1)(x-5))$ using partial fractions?
How do you integrate $(2x-3)/(x^2-5x+6)$ using partial fractions?
How do you integrate $(5x)/(2x^2+11x+12)$ using partial fractions?
How do you integrate $(5x+1)/((x+3)(x+2)(x-4))$ using partial fractions?
How do you integrate $(4x^2-x+3)/((x+5)(x-1)(x-2))$ using partial fractions?
How do you integrate $(2x^2-5)/(x^3-2x^2-3x)$ using partial fractions?
How do you integrate $(x+2)/(x^2+8x+16)$ using partial fractions?
How do you integrate $(5-4x)/(x^3+10x^2+25x)$ using partial fractions?
How do you integrate $(5x^2-9x)/((x-4)(x-1)^2)$ using partial fractions?
How do you integrate $(x^2-2)/((x+1)(x^2+3))$ using partial fractions?
How do you integrate $(3x^2+9x-4)/((x-1)(x^2+4x-1))$ using partial fractions?
How do you integrate $6/((x-5)(x^2-2x+3))$ using partial fractions?
How do you integrate $x^3/(x^2+2x+1)$ using partial fractions?
How do you integrate $(2y^3+5y^2-8y-4)/(y^2+2y+2)$ using partial fractions?
How do you integrate $(2x)/((x-1)(x+1))$ using partial fractions?
How do you integrate $(2x)/((x^2+x+1)(x-1))$ using partial fractions?
How do you integrate $(x^2-1)/(x^4-16)$ using partial fractions?
How do you integrate $(x+2)/(2x^3-8x)$ using partial fractions?
How do you integrate $x^2/((x-1)(x^2+2x+1))$ using partial fractions?
How do you integrate $(y^4+y^2-1)/(y^3+y)$ using partial fractions?
How do you integrate $(2y^4)/(y^3-y^2+y-1)$ using partial fractions?
How do you integrate $1/(x-3)$ using partial fractions?
How do you integrate $7/(2x-3)$ using partial fractions?
How do you integrate $7/(x-3)^2$ using partial fractions?
How do you integrate $7/(x^2+1)$ using partial fractions?
How do you integrate $(7+8x)/(x^2+1)$ using partial fractions?
How do you integrate $8/((x-3)^2+1)$ using partial fractions?
How do you integrate $8/(x^2-8x+17)$ using partial fractions?
How do you find the indefinite integral of $int 5/x$ ?
How do you find the indefinite integral of $int 10/x$ ?
How do you find the indefinite integral of $int 1/(x+1)$ ?
How do you find the indefinite integral of $int 1/(x-5)$ ?
How do you find the indefinite integral of $int 1/(3-2x)$ ?
How do you find the indefinite integral of $int 1/(3x+2)$ ?
How do you find the indefinite integral of $int x/(x^2+1)$ ?
How do you find the indefinite integral of $int x^2/(3-x^2)$ ?
How do you find the indefinite integral of $int (x^2-4)/x$ ?
How do you find the indefinite integral of $int x/(sqrt(9-x^2))$ ?
How do you find the indefinite integral of $int (x^2+2x+3)/(x^2+3x^2+9x)$ ?
How do you find the indefinite integral of $int (x(x+2))/(x^3+3x^2-4)$ ?
How do you find the indefinite integral of $int (x^2-3x+2)/(x-1) dx$ ?
How do you find the indefinite integral of $int (2x^2+7x-3)/(x-2)$ ?
How do you find the indefinite integral of $int (x^2-3x^2+5)/(x-3)$ ?
How do you find the indefinite integral of $int (x^2-6x-20)/(x+5)$ ?
How do you find the indefinite integral of $int (x^4+x-4)/(x^2+2)$ ?
How do you find the indefinite integral of $int (x^2-6x-20)/(x+5)$ ?
How do you find the indefinite integral of $int (x^4+x-4)/(x^2+2)$ ?
How do you find the indefinite integral of $int (x^3-3x^2+4x-9)/(x^2+3)$ ?
How do you find the indefinite integral of $int (lnx)^2/x$ ?
How do you find the indefinite integral of $int 1/(xlnx^3)$ ?
How do you find the indefinite integral of $int 1/sqrt(x+1)$ ?
How do you find the indefinite integral of $int 1/(x^(2/3)(1+x^(1/3))$ ?
How do you find the indefinite integral of $int (2x)/(x-1)^2$ ?
How do you find the indefinite integral of $int (x(x-2))/(x-1)^3$ ?
How do you find the definite integral of $int 5/(3x+1)$ from [0,4]?
How do you find the definite integral of $int 1/(x+2)$ from [-1,1]?
How do you find the definite integral of $int 1/(xlnx)$ from $[e,e^2]$ ?
How do you find the definite integral of $int (x^2-2)/(x+1)$ from $[0,2]$ ?
How do you find the definite integral of $int (1-costheta)/(theta-sintheta)$ from $[1,2]$ ?
How do you find b such that $intdx/(1+x^2)=pi/3$ from [0,b]?
How do I find the anti derivative of $(2x^2+5)/(x^2+1)$ ?
How do you evaluate the integral $intdx/(2x+3)$ ?
How do you evaluate the integral $int (xdx)/(5x^2-2)$ ?
How do you evaluate the integral $int (2xdx)/(x-1)$ ?
How do you evaluate the integral $int (x-1)/(x+1)dx$ ?
How do you evaluate the integral $int lnx/xdx$ ?
How do you evaluate the integral $int e^t/(e^t+1)dt$ ?
How do you evaluate the integral $int dt/(tlnt)$ ?
How do you evaluate the integral $int costheta/(1+sintheta)$ ?
How do you evaluate the integral $int (ln(lnx))/x dx$ ?
How do you evaluate the integral $int xlnxdx$ ?
How do you evaluate the integral $intx^nlnxdx$ ?
How do you evaluate the integral $int (lnx)^2dx$ ?
How do you evaluate the integral $int (lnx)^3dx$ ?
How do you evaluate the integral $int x(lnx)^2dx$ ?
How do you evaluate the integral $int 1/xcos(lnx)dx$ ?
How do you evaluate the integral $int cos(lnx)dx$ ?
How do you evaluate the integral $int x^4/(1+x^2)dx$ ?
How do you evaluate the integral $int x^2/(4+x^2)^2dx$ ?
How do you evaluate the integral $int 1/((1+x^2)(2+x^2))dx$ ?
How do you evaluate the integral $int (x^4+3x+1)/(x^2+x+1)dx$ ?
How do you evaluate the integral $int dx/(x^4+x^2)$ ?
How do you evaluate the integral $int dx/(x^4-16)$ ?
How do you evaluate the integral $int dx/(x^3+1)$ ?
How do you evaluate the integral $int (3x+6)/(x^4-2x^2+1)$ ?
How do you evaluate the integral $int (x^5+3x^2+1)/(x^4-1)dx$ ?
How do you evaluate the integral $int dx/(x^4+1)$ ?
How do you evaluate the integral $int x/(root3(x^2-1))$ ?
How do you integrate $int (x^4+1)/(x^5+5x+3)$ ?
How do you integrate $int e^(5x)cos3x$ ?
Integration of $(x-1)/(sqrtx(x-1))$ dx?
How to integrate this? $intcolor(white).(3x^2-x+1)/(x^3-x^2)$

Introduction to Integration

View all chapters

Sigma Notation
Integration: the Area Problem
Formal Definition of the Definite Integral
Definite and indefinite integrals
Integrals of Polynomial functions
Determining Basic Rates of Change Using Integrals
Integrals of Trigonometric Functions
Integrals of Exponential Functions
Integrals of Rational Functions
The Fundamental Theorem of Calculus
Basic Properties of Definite Integrals

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Integrals of Rational Functions

Key Questions

Answer:

Explanation:

Questions

Introduction to Integration