integral of x^45^x

解

\int x^{4} 5^{x} d x

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

解答ステップ

\int x^{4} \cdot 5^{x} d x

部分積分を適用する

= \frac{5 ^{x} x ^{4}}{ln ( 5 )} - \int \frac{4 \cdot 5 ^{x} x ^{3}}{ln ( 5 )} d x

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

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

定数を解答に追加する

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