integral from 0 to 5 of 5^{5x}

解

\int_{0}^{5} 5^{5 x} d x

\frac{5 ^{24}}{ln ( 5 )} - \frac{1}{5 ln ( 5 )}

十進法表記

3.70344 E 16

解答ステップ

\int_{0}^{5} 5^{5 x} d x

簡素化

5^{5 x} : 312 5^{x}

= \int_{0}^{5} 312 5^{x} d x

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

= [\frac{312 5 ^{x}}{ln ( 3125 )}]_{0}^{5}

\frac{312 5 ^{x}}{ln ( 3125 )} = \frac{5 ^{5 x - 1}}{ln ( 5 )}

= [\frac{5 ^{5 x - 1}}{ln ( 5 )}]_{0}^{5}

限度を計算する:

\frac{5 ^{24}}{ln ( 5 )} - \frac{1}{5 ln ( 5 )}

= \frac{5 ^{24}}{ln ( 5 )} - \frac{1}{5 ln ( 5 )}