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

$\int \frac{x^3+1}{x^2+4}\ dx$

$=\int(x-\frac{4x}{x^2+4}+\frac{1}{x^2+4})\ dx$

$=\int\ x\ dx-\int \frac{4x}{x^2+4}\ dx+\int \frac{1}{x^2+4}\ dx$

$=x^2/2-2\int \frac{d(x^2+4)}{x^2+4}\ dx+\int \frac{1}{x^2+2^2}\ dx$

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

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

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