How do you integrate #int (x^3+1) / (x^2+4)# using partial fractions?

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Answer 1 · 2018-07-27T12:46:19

#x^2/2+1/{2}\tan^{-1}(x/2)-2\ln(x^2+4)+C#

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#