integral of x^2a^{2x}

解

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

\frac{a ^{2 x} x ^{2}}{2 ln ( a )} - \frac{1}{ln ( a )} (\frac{a ^{2 x} x}{2 ln ( a )} - \frac{a ^{2 x}}{4 ln ^{2} ( a )}) + C

解答ステップ

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

部分積分を適用する

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

\int \frac{a ^{2 x} x}{ln ( a )} d x = \frac{1}{ln ( a )} (\frac{a ^{2 x} x}{2 ln ( a )} - \frac{a ^{2 x}}{4 ln ^{2} ( a )})

= \frac{a ^{2 x} x ^{2}}{2 ln ( a )} - \frac{1}{ln ( a )} (\frac{a ^{2 x} x}{2 ln ( a )} - \frac{a ^{2 x}}{4 ln ^{2} ( a )})

定数を解答に追加する

= \frac{a ^{2 x} x ^{2}}{2 ln ( a )} - \frac{1}{ln ( a )} (\frac{a ^{2 x} x}{2 ln ( a )} - \frac{a ^{2 x}}{4 ln ^{2} ( a )}) + C