integral from 1 to e^2 of (ln(z))/(xz)

解

\int_{1}^{e^{2}} \frac{ln ( z )}{x z} d z

\frac{2}{x}

解答ステップ

\int_{1}^{e^{2}} \frac{ln ( z )}{x z} d z

定数を除く:

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

= \frac{1}{x} \cdot \int_{1}^{e^{2}} \frac{ln ( z )}{z} d z

u置換積分法を適用する

= \frac{1}{x} \cdot \int_{0}^{2} u d u

べき乗則を適用する

= \frac{1}{x} [\frac{u ^{2}}{2}]_{0}^{2}

簡素化

\frac{1}{x} [\frac{u ^{2}}{2}]_{0}^{2} : \frac{[ \frac{u ^{2}}{2} ] _{0}^{2}}{x}

= \frac{[ \frac{u ^{2}}{2} ] _{0}^{2}}{x}

限度を計算する:

2

= \frac{2}{x}