integral of x^23^x

解

\int x^{2} 3^{x} d x

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

解答ステップ

\int x^{2} \cdot 3^{x} d x

部分積分を適用する

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

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

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

定数を解答に追加する

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