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人気のある微分積分 >

integral from 0 to 1 of 3x5^x

解

\int_{0}^{1} 3 x 5^{x} d x

解

\frac{3 ( 5 ln ( 5 ) - 4 )}{ln ^{2} ( 5 )}

+1

十進法表記

4.68733 \dots

解答ステップ

\int_{0}^{1} 3 x \cdot 5^{x} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int_{0}^{1} x \cdot 5^{x} d x

部分積分を適用する

= 3 [\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 )}

= 3 [\frac{5 ^{x} x}{ln ( 5 )} - \frac{5 ^{x}}{ln ^{2} ( 5 )}]_{0}^{1}

限度を計算する:

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

= 3 \cdot \frac{5 ln ( 5 ) - 4}{ln ^{2} ( 5 )}

簡素化

= \frac{3 ( 5 ln ( 5 ) - 4 )}{ln ^{2} ( 5 )}

グラフ

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