integral from 1 to 2 of (3^{2ln(x)})/x

解

\int_{1}^{2} \frac{3 ^{2 l n (x)}}{x} d x

\frac{9 ^{l n (2)} - 1}{2 ln ( 3 )}

十進法表記

1.63204 \dots

解答ステップ

\int_{1}^{2} \frac{3 ^{2 l n (x)}}{x} d x

3^{2 l n (x)} = 9^{l n (x)}

= \int_{1}^{2} \frac{9 ^{l n (x)}}{x} d x

u置換積分法を適用する

= \int_{0}^{l n (2)} 9^{u} d u

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

= [\frac{9 ^{u}}{ln ( 9 )}]_{0}^{l n (2)}

簡素化

ln (9) : 2 ln (3)

= [\frac{9 ^{u}}{2 ln ( 3 )}]_{0}^{l n (2)}

限度を計算する:

\frac{9 ^{l n (2)} - 1}{2 ln ( 3 )}

= \frac{9 ^{l n (2)} - 1}{2 ln ( 3 )}