integral from 0 to 1 of x7^x

解

\int_{0}^{1} x 7^{x} d x

\frac{7 ln ( 7 ) - 6}{ln ^{2} ( 7 )}

十進法表記

2.01273 \dots

解答ステップ

\int_{0}^{1} x \cdot 7^{x} d x

部分積分を適用する

= [\frac{7 ^{x} x}{ln ( 7 )} - \int \frac{7 ^{x}}{ln ( 7 )} d x]_{0}^{1}

\int \frac{7 ^{x}}{ln ( 7 )} d x = \frac{7 ^{x}}{ln ^{2} ( 7 )}

= [\frac{7 ^{x} x}{ln ( 7 )} - \frac{7 ^{x}}{ln ^{2} ( 7 )}]_{0}^{1}

限度を計算する:

\frac{7 ln ( 7 ) - 6}{ln ^{2} ( 7 )}

= \frac{7 ln ( 7 ) - 6}{ln ^{2} ( 7 )}