What is the integral from 0 to x of (e^{2s}-1)/s ?

The integral from 0 to x of (e^{2s}-1)/s is diverges

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integral from 0 to x of (e^{2s}-1)/s

\int_{0}^{x} \frac{e ^{2 s} - 1}{s} d s

diverges

Solution steps

\int_{0}^{x} \frac{e ^{2 s} - 1}{s} d s

Expand

\frac{e ^{2 s} - 1}{s} : \frac{e ^{2 s}}{s} - \frac{1}{s}

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

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}^{x} \frac{e ^{2 s}}{s} d s - \int_{0}^{x} \frac{1}{s} d s

\int_{0}^{x} \frac{e ^{2 s}}{s} d s = Ei (2 x) - Ei (0)

\int_{0}^{x} \frac{1}{s} d s =

diverges

If part of the expression diverges, then entire expression diverges

= diverges

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What is the integral from 0 to x of (e^{2s}-1)/s ?
The integral from 0 to x of (e^{2s}-1)/s is diverges