integral from 0 to ln(a) of 9e^{3x}

解

\int_{0}^{l n (a)} 9 e^{3 x} d x

3 (a^{3} - 1)

解答ステップ

\int_{0}^{l n (a)} 9 e^{3 x} d x

定数を除く:

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

= 9 \cdot \int_{0}^{l n (a)} e^{3 x} d x

u置換積分法を適用する

= 9 \cdot \int_{0}^{3 l n (a)} e^{u} \frac{1}{3} d u

定数を除く:

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

= 9 \cdot \frac{1}{3} \cdot \int_{0}^{3 l n (a)} e^{u} d u

共通積分を使用する:

\int e^{u} d u = e^{u}

= 9 \cdot \frac{1}{3} [e^{u}]_{0}^{3 l n (a)}

簡素化

9 \cdot \frac{1}{3} [e^{u}]_{0}^{3 l n (a)} : 3 [e^{u}]_{0}^{3 l n (a)}

= 3 [e^{u}]_{0}^{3 l n (a)}

限度を計算する:

a^{3} - 1

= 3 (a^{3} - 1)