integral from 0 to 1 of x5^x

解答

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

\frac{5 ln ( 5 ) - 4}{ln ^{2} ( 5 )}

十进制

1.56244 \dots

求解步骤

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

使用分布积分法

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

\int \frac{5 ^{x}}{ln ( 5 )} d x = \frac{5 ^{x}}{ln ^{2} ( 5 )}

= [\frac{5 ^{x} x}{ln ( 5 )} - \frac{5 ^{x}}{ln ^{2} ( 5 )}]_{0}^{1}

带入上下限计算后得:

\frac{5 ln ( 5 ) - 4}{ln ^{2} ( 5 )}

= \frac{5 ln ( 5 ) - 4}{ln ^{2} ( 5 )}

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