integral from 0 to 1 of (3^x)/(e^x)

解

\int_{0}^{1} \frac{3 ^{x}}{e ^{x}} d x

\frac{3}{e ( ln ( 3 ) - 1 )} - \frac{1}{ln ( 3 ) - 1}

十進法表記

1.05096 \dots

解答ステップ

\int_{0}^{1} \frac{3 ^{x}}{e ^{x}} d x

指数の規則を適用する:

\frac{a ^{c}}{b ^{c}} = (\frac{a}{b})^{c}

= \int_{0}^{1} (\frac{3}{e})^{x} d x

\int a^{x} d x = \frac{a ^{x}}{ln a}

= [\frac{( \frac{3}{e} ) ^{x}}{ln ( \frac{3}{e} )}]_{0}^{1}

簡素化

[\frac{( \frac{3}{e} ) ^{x}}{ln ( \frac{3}{e} )}]_{0}^{1} : [\frac{3 ^{x}}{( ln ( 3 ) - 1 ) e ^{x}}]_{0}^{1}

= [\frac{3 ^{x}}{( ln ( 3 ) - 1 ) e ^{x}}]_{0}^{1}

限度を計算する:

\frac{3}{e ( ln ( 3 ) - 1 )} - \frac{1}{ln ( 3 ) - 1}

= \frac{3}{e ( ln ( 3 ) - 1 )} - \frac{1}{ln ( 3 ) - 1}