integral from 0 to 2 of a^{x/2}

解

\int_{0}^{2} a^{\frac{x}{2}} d x

\frac{2 ( a - 1 )}{ln ( a )}

解答ステップ

\int_{0}^{2} a^{\frac{x}{2}} d x

u置換積分法を適用する

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

定数を除く:

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

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

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

= 2 [\frac{a ^{u}}{ln ( a )}]_{0}^{1}

限度を計算する:

\frac{a}{ln ( a )} - \frac{1}{ln ( a )}

= 2 (\frac{a}{ln ( a )} - \frac{1}{ln ( a )})

簡素化

= \frac{2 ( a - 1 )}{ln ( a )}