integral of ((2+ln(x^2)))/x

解

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

2 ln ∣ x ∣ + ln^{2} (x) + C

解答ステップ

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

拡張

\frac{2 + ln ( x ^{2} )}{x} : \frac{2}{x} + \frac{ln ( x ^{2} )}{x}

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

= 2 ln ∣ x ∣ + ln^{2} (x)

定数を解答に追加する

= 2 ln ∣ x ∣ + ln^{2} (x) + C