integral of (x+5)^9

解

\int (x + 5)^{9} d x

x (x + 5)^{9} - \frac{9}{10} (x + 5)^{10} + 5 (x + 5)^{9} + C

解答ステップ

\int (x + 5)^{9} d x

部分積分を適用する

= (x + 5)^{9} x - \int 9 (x + 5)^{8} x d x

\int 9 (x + 5)^{8} x d x = \frac{9}{10} (x + 5)^{10} - 5 (x + 5)^{9}

= (x + 5)^{9} x - (\frac{9}{10} (x + 5)^{10} - 5 (x + 5)^{9})

簡素化

(x + 5)^{9} x - (\frac{9}{10} (x + 5)^{10} - 5 (x + 5)^{9}) : x (x + 5)^{9} - \frac{9}{10} (x + 5)^{10} + 5 (x + 5)^{9}

= x (x + 5)^{9} - \frac{9}{10} (x + 5)^{10} + 5 (x + 5)^{9}

定数を解答に追加する

= x (x + 5)^{9} - \frac{9}{10} (x + 5)^{10} + 5 (x + 5)^{9} + C