What is the integral of ((x^3+4))/(x(x^2+2)) ?

The integral of ((x^3+4))/(x(x^2+2)) is -sqrt(2)arctan(x/(sqrt(2)))+x-ln|x^2+2|+2ln(x)+C

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integral of ((x^3+4))/(x(x^2+2))

\int \frac{( x ^{3} + 4 )}{x ( x ^{2} + 2 )} d x

- 2 arctan (\frac{x}{2}) + x - ln x^{2} + 2 + 2 ln (x) + C

Solution steps

\int \frac{x ^{3} + 4}{x ( x ^{2} + 2 )} d x

Expand

\frac{x ^{3} + 4}{x ( x ^{2} + 2 )} : \frac{x ^{3}}{x ( x ^{2} + 2 )} + \frac{4}{x ( x ^{2} + 2 )}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{3}}{x ( x ^{2} + 2 )} d x + \int \frac{4}{x ( x ^{2} + 2 )} d x

\int \frac{x ^{3}}{x ( x ^{2} + 2 )} d x = - 2 arctan (\frac{x}{2}) + x

\int \frac{4}{x ( x ^{2} + 2 )} d x = - ln x^{2} + 2 + 2 ln (x)

= - 2 arctan (\frac{x}{2}) + x - ln x^{2} + 2 + 2 ln (x)

Add a constant to the solution

= - 2 arctan (\frac{x}{2}) + x - ln x^{2} + 2 + 2 ln (x) + C

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What is the integral of ((x^3+4))/(x(x^2+2)) ?
The integral of ((x^3+4))/(x(x^2+2)) is -sqrt(2)arctan(x/(sqrt(2)))+x-ln|x^2+2|+2ln(x)+C