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

解

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

e - \frac{π}{2} erfi (ln (e))

解答ステップ

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

部分積分を適用する

= [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}

簡素化

[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}

限度を計算する:

e - \frac{π}{2} erfi (ln (e))

= e - \frac{π}{2} erfi (ln (e))