integral of [x-5/((3x+5)^2)]

解

\int [x - \frac{5}{( 3 x + 5 ) ^{2}}] d x

\frac{x ^{2}}{2} + \frac{5}{3 ( 3 x + 5 )} + C

解答ステップ

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

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int x d x = \frac{x ^{2}}{2}

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

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

簡素化

= \frac{x ^{2}}{2} + \frac{5}{3 ( 3 x + 5 )}

定数を解答に追加する

= \frac{x ^{2}}{2} + \frac{5}{3 ( 3 x + 5 )} + C