integral of (ln(5x)+5)/x

解

\int \frac{ln ( 5 x ) + 5}{x} d x

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

解答ステップ

\int \frac{ln ( 5 x ) + 5}{x} d x

拡張

\frac{ln ( 5 x ) + 5}{x} : \frac{ln ( 5 x )}{x} + \frac{5}{x}

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

\int \frac{ln ( 5 x )}{x} d x = \frac{ln ^{2} ( 5 x )}{2}

\int \frac{5}{x} d x = 5 ln ∣ x ∣

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

定数を解答に追加する

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