What is the integral from 0 to 1 of 1/(x(x^2+1)) ?

The integral from 0 to 1 of 1/(x(x^2+1)) is diverges

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integral from 0 to 1 of 1/(x(x^2+1))

\int_{0}^{1} \frac{1}{x ( x ^{2} + 1 )} d x

diverges

Solution steps

\int_{0}^{1} \frac{1}{x ( x ^{2} + 1 )} d x

Take the partial fraction of

\frac{1}{x ( x ^{2} + 1 )} : \frac{1}{x} - \frac{x}{x ^{2} + 1}

= \int_{0}^{1} \frac{1}{x} - \frac{x}{x ^{2} + 1} 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}^{1} \frac{1}{x} d x - \int_{0}^{1} \frac{x}{x ^{2} + 1} d x

\int_{0}^{1} \frac{1}{x} 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 1/(x(x^2+1)) ?
The integral from 0 to 1 of 1/(x(x^2+1)) is diverges