integral of x(3x+5)^2

解

\int x (3 x + 5)^{2} d x

\frac{9}{2} x^{4} - \frac{9 x ^{4}}{4} + 10 x^{3} + \frac{25}{2} x^{2} + C

解答ステップ

\int x (3 x + 5)^{2} d x

部分積分を適用する

= \frac{1}{2} x^{2} (3 x + 5)^{2} - \int 3 x^{2} (3 x + 5) d x

\int 3 x^{2} (3 x + 5) d x = \frac{9 x ^{4}}{4} + 5 x^{3}

= \frac{1}{2} x^{2} (3 x + 5)^{2} - (\frac{9 x ^{4}}{4} + 5 x^{3})

簡素化

\frac{1}{2} x^{2} (3 x + 5)^{2} - (\frac{9 x ^{4}}{4} + 5 x^{3}) : \frac{9}{2} x^{4} - \frac{9 x ^{4}}{4} + 10 x^{3} + \frac{25}{2} x^{2}

= \frac{9}{2} x^{4} - \frac{9 x ^{4}}{4} + 10 x^{3} + \frac{25}{2} x^{2}

定数を解答に追加する

= \frac{9}{2} x^{4} - \frac{9 x ^{4}}{4} + 10 x^{3} + \frac{25}{2} x^{2} + C