integral of x^33^x

解

\int x^{3} 3^{x} d x

\frac{3 ^{x} x ^{3}}{ln ( 3 )} - \frac{1}{ln ( 3 )} (\frac{3 ^{x + 1} x ^{2}}{ln ( 3 )} - \frac{2}{ln ( 3 )} (\frac{3 ^{x + 1} x}{ln ( 3 )} - \frac{1}{ln ^{2} ( 3 )} \cdot 3^{1 + x})) + C

解答ステップ

\int x^{3} \cdot 3^{x} d x

部分積分を適用する

= \frac{3 ^{x} x ^{3}}{ln ( 3 )} - \int \frac{3 ^{x + 1} x ^{2}}{ln ( 3 )} d x

\int \frac{3 ^{x + 1} x ^{2}}{ln ( 3 )} d x = \frac{1}{ln ( 3 )} (\frac{3 ^{x + 1} x ^{2}}{ln ( 3 )} - \frac{2}{ln ( 3 )} (\frac{3 ^{x + 1} x}{ln ( 3 )} - \frac{1}{ln ^{2} ( 3 )} \cdot 3^{1 + x}))

= \frac{3 ^{x} x ^{3}}{ln ( 3 )} - \frac{1}{ln ( 3 )} (\frac{3 ^{x + 1} x ^{2}}{ln ( 3 )} - \frac{2}{ln ( 3 )} (\frac{3 ^{x + 1} x}{ln ( 3 )} - \frac{1}{ln ^{2} ( 3 )} \cdot 3^{1 + x}))

定数を解答に追加する

= \frac{3 ^{x} x ^{3}}{ln ( 3 )} - \frac{1}{ln ( 3 )} (\frac{3 ^{x + 1} x ^{2}}{ln ( 3 )} - \frac{2}{ln ( 3 )} (\frac{3 ^{x + 1} x}{ln ( 3 )} - \frac{1}{ln ^{2} ( 3 )} \cdot 3^{1 + x})) + C