How do you integrate #(3x) / (x^2 * (x^2+1) )# using partial fractions? Calculus Techniques of Integration Integral by Partial Fractions 1 Answer Ratnaker Mehta Jan 20, 2017 #3/2ln{x^2/(x^2+1)}+C.# Explanation: Let #I=int(3x)/{x^2(x^2+1)}dx# Substitute #x^2=t," so that, "2xdx=dt, or, xdx=1/2dt# #:. I=3int(1/2)/{t(t+1)}dt=3/2int{(t+1)-t}/{t(t+1)}dt# #=3/2int[(t+1)/{t(t+1)}-t/{t(t+1)}]dt# #=3/2int[1/t-1/(t+1)]dt# #=3/2[ln|t|-ln|t+1|]# #=3/2ln|t/(t+1)|, and, because, t=x^2,# #I=3/2ln{x^2/(x^2+1)}+C.# Enjoy Maths.! Answer link Related questions How do I find the partial fraction decomposition of #(2x)/((x+3)(3x+1))# ? How do I find the partial fraction decomposition of #(1)/(x^3+2x^2+x# ? How do I find the partial fraction decomposition of #(x^4+1)/(x^5+4x^3)# ? How do I find the partial fraction decomposition of #(x^4)/(x^4-1)# ? How do I find the partial fraction decomposition of #(t^4+t^2+1)/((t^2+1)(t^2+4)^2)# ? How do I find the integral #intt^2/(t+4)dt# ? How do I find the integral #int(x-9)/((x+5)(x-2))dx# ? How do I find the integral #int1/((w-4)(w+1))dw# ? How do I find the integral #intdx/(x^2(x-1)^2)# ? How do I find the integral #int(x^3+4)/(x^2+4)dx# ? See all questions in Integral by Partial Fractions Impact of this question 1165 views around the world You can reuse this answer Creative Commons License