What is the integral from 1 to e of sqrt(ln(x)) ?

The integral from 1 to e of sqrt(ln(x)) is e-(sqrt(pi))/2 \erfi(sqrt(ln(e)))

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integral from 1 to e of sqrt(ln(x))

\int_{1}^{e} ln (x) d x

e - \frac{π}{2} erfi (ln (e))

Solution steps

\int_{1}^{e} ln (x) d x

Apply Integration By Parts

= [x ln (x) - \int \frac{1}{2 ln ( x )} d x]_{1}^{e}

\int \frac{1}{2 ln ( x )} d x = \frac{π}{2} erfi (ln (x))

= [x ln (x) - \frac{π}{2} erfi (ln (x))]_{1}^{e}

Simplify

[x ln (x) - \frac{π}{2} erfi (ln (x))]_{1}^{e} : [\frac{1}{2} (2 x ln (x) - π erfi (ln (x)))]_{1}^{e}

= [\frac{1}{2} (2 x ln (x) - π erfi (ln (x)))]_{1}^{e}

Compute the boundaries:

e - \frac{π}{2} erfi (ln (e))

= e - \frac{π}{2} erfi (ln (e))

\int \frac{x}{3 - x} d x

\frac{e ^{x}}{3 + e ^{x}}

\frac{d}{d x} (9 x^{5} cos (x))

\frac{d}{d x} (x + x^{5})

\frac{1}{1 + x ^{4}}

What is the integral from 1 to e of sqrt(ln(x)) ?
The integral from 1 to e of sqrt(ln(x)) is e-(sqrt(pi))/2 \erfi(sqrt(ln(e)))