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

• 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)#