integral from 0 to 1 of 5^{5x}

解答

\int_{0}^{1} 5^{5 x} d x

\frac{3124}{5 ln ( 5 )}

十进制

388.21006 \dots

求解步骤

\int_{0}^{1} 5^{5 x} d x

化简

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

= \int_{0}^{1} 312 5^{x} d x

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

= [\frac{312 5 ^{x}}{ln ( 3125 )}]_{0}^{1}

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

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

带入上下限计算后得:

\frac{3124}{5 ln ( 5 )}

= \frac{3124}{5 ln ( 5 )}

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