How do you integrate #int (x^3+1) / (x^2+4)# using partial fractions? Calculus Techniques of Integration Integral by Partial Fractions 1 Answer Harish Chandra Rajpoot Jul 27, 2018 #x^2/2+1/{2}\tan^{-1}(x/2)-2\ln(x^2+4)+C# Explanation: Given that #\int \frac{x^3+1}{x^2+4}\ dx# #=\int(x-\frac{4x}{x^2+4}+\frac{1}{x^2+4})\ dx# #=\int\ x\ dx-\int \frac{4x}{x^2+4}\ dx+\int \frac{1}{x^2+4}\ dx# #=x^2/2-2\int \frac{d(x^2+4)}{x^2+4}\ dx+\int \frac{1}{x^2+2^2}\ dx# #=x^2/2-2\ln|x^2+4|+1/{2}\tan^{-1}(x/2)+C# #=x^2/2+1/{2}\tan^{-1}(x/2)-2\ln(x^2+4)+C# 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 1144 views around the world You can reuse this answer Creative Commons License