What is the integral from 0 to 5 of 1/(125-x^3) ?

The integral from 0 to 5 of 1/(125-x^3) is diverges

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integral from 0 to 5 of 1/(125-x^3)

\int_{0}^{5} \frac{1}{125 - x ^{3}} d x

diverges

Solution steps

\int_{0}^{5} \frac{1}{125 - x ^{3}} d x

Factor

125 - x^{3} : - (- 125 + x^{3})

= \int_{0}^{5} \frac{1}{- ( - 125 + x ^{3} )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int_{0}^{5} \frac{1}{- 125 + x ^{3}} d x

Take the partial fraction of

\frac{1}{- 125 + x ^{3}} : \frac{1}{75 ( x - 5 )} + \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )}

= - \int_{0}^{5} \frac{1}{75 ( x - 5 )} + \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )} d x

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}^{5} \frac{1}{75 ( x - 5 )} d x + \int_{0}^{5} \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )} d x)

\int_{0}^{5} \frac{1}{75 ( x - 5 )} 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 5 of 1/(125-x^3) ?
The integral from 0 to 5 of 1/(125-x^3) is diverges