integral of (x^2)/((5x+7)^3)

解

\int \frac{x ^{2}}{( 5 x + 7 ) ^{3}} d x

- \frac{x ^{2}}{10 ( 5 x + 7 ) ^{2}} + \frac{1}{5} (- \frac{x}{5 ( 5 x + 7 )} + \frac{1}{25} ln ∣ 5 x + 7 ∣) + C

解答ステップ

\int \frac{x ^{2}}{( 5 x + 7 ) ^{3}} d x

部分積分を適用する

= - \frac{x ^{2}}{10 ( 5 x + 7 ) ^{2}} - \int - \frac{x}{5 ( 5 x + 7 ) ^{2}} d x

\int - \frac{x}{5 ( 5 x + 7 ) ^{2}} d x = - \frac{1}{5} (- \frac{x}{5 ( 5 x + 7 )} + \frac{1}{25} ln ∣ 5 x + 7 ∣)

= - \frac{x ^{2}}{10 ( 5 x + 7 ) ^{2}} - (- \frac{1}{5} (- \frac{x}{5 ( 5 x + 7 )} + \frac{1}{25} ln ∣ 5 x + 7 ∣))

簡素化

= - \frac{x ^{2}}{10 ( 5 x + 7 ) ^{2}} + \frac{1}{5} (- \frac{x}{5 ( 5 x + 7 )} + \frac{1}{25} ln ∣ 5 x + 7 ∣)

定数を解答に追加する

= - \frac{x ^{2}}{10 ( 5 x + 7 ) ^{2}} + \frac{1}{5} (- \frac{x}{5 ( 5 x + 7 )} + \frac{1}{25} ln ∣ 5 x + 7 ∣) + C