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

解

\int \frac{2 x}{5 x + 3} d x

2 (\frac{1}{5} x - \frac{3}{25} ln ∣ 5 x + 3 ∣) + C

解答ステップ

\int \frac{2 x}{5 x + 3} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{x}{5 x + 3} d x

長除法

\frac{x}{5 x + 3} : \frac{1}{5} - \frac{3}{5 ( 5 x + 3 )}

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

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

\int \frac{1}{5} d x = \frac{1}{5} x

\int \frac{3}{5 ( 5 x + 3 )} d x = \frac{3}{25} ln ∣ 5 x + 3 ∣

= 2 (\frac{1}{5} x - \frac{3}{25} ln ∣ 5 x + 3 ∣)

定数を解答に追加する

= 2 (\frac{1}{5} x - \frac{3}{25} ln ∣ 5 x + 3 ∣) + C