integral of (ln(x)+1)/(xln(x))

解

\int \frac{ln ( x ) + 1}{x ln ( x )} d x

ln ∣ x ∣ + ln ∣ ln (x) ∣ + C

解答ステップ

\int \frac{ln ( x ) + 1}{x ln ( x )} d x

拡張

\frac{ln ( x ) + 1}{x ln ( x )} : \frac{1}{x} + \frac{1}{x ln ( x )}

= \int \frac{1}{x} + \frac{1}{x ln ( 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{1}{x} d x + \int \frac{1}{x ln ( x )} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{1}{x ln ( x )} d x = ln ∣ ln (x) ∣

= ln ∣ x ∣ + ln ∣ ln (x) ∣

定数を解答に追加する

= ln ∣ x ∣ + ln ∣ ln (x) ∣ + C