What is the integral from 0 to 1 of (x^3+4)/(x(x+1)) ?

The integral from 0 to 1 of (x^3+4)/(x(x+1)) is diverges

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integral from 0 to 1 of (x^3+4)/(x(x+1))

\int_{0}^{1} \frac{x ^{3} + 4}{x ( x + 1 )} d x

diverges

Solution steps

\int_{0}^{1} \frac{x ^{3} + 4}{x ( x + 1 )} d x

Expand

\frac{x ^{3} + 4}{x ( x + 1 )} : \frac{x ^{3}}{x ( x + 1 )} + \frac{4}{x ( x + 1 )}

Apply the Sum Rule:

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

= \int_{0}^{1} \frac{x ^{3}}{x ( x + 1 )} d x + \int_{0}^{1} \frac{4}{x ( x + 1 )} d x

\int_{0}^{1} \frac{x ^{3}}{x ( x + 1 )} d x = ln (2) - \frac{1}{2}

\int_{0}^{1} \frac{4}{x ( x + 1 )} d x =

diverges

If part of the expression diverges, then entire expression diverges

= diverges

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What is the integral from 0 to 1 of (x^3+4)/(x(x+1)) ?
The integral from 0 to 1 of (x^3+4)/(x(x+1)) is diverges