integral of (u^2)/((a+bu)^2)

解

\int \frac{u ^{2}}{( a + b u ) ^{2}} d u

- \frac{u ^{2}}{b ( a + b u )} + \frac{2}{b ^{3}} (a + b u - a ln ∣ a + b u ∣) + C

解答ステップ

\int \frac{u ^{2}}{( a + b u ) ^{2}} d u

部分積分を適用する

= - \frac{u ^{2}}{b ( a + b u )} - \int - \frac{2 u}{b ( a + b u )} d u

\int - \frac{2 u}{b ( a + b u )} d u = - \frac{2}{b ^{3}} (a + b u - a ln ∣ a + b u ∣)

= - \frac{u ^{2}}{b ( a + b u )} - (- \frac{2}{b ^{3}} (a + b u - a ln ∣ a + b u ∣))

簡素化

= - \frac{u ^{2}}{b ( a + b u )} + \frac{2}{b ^{3}} (a + b u - a ln ∣ a + b u ∣)

定数を解答に追加する

= - \frac{u ^{2}}{b ( a + b u )} + \frac{2}{b ^{3}} (a + b u - a ln ∣ a + b u ∣) + C