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

解

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

x ln (ln (x)) + C

解答ステップ

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

\int ln (ln (x)) d x = x ln (ln (x)) - Ei (ln (x))

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

= x ln (ln (x)) - Ei (ln (x)) + Ei (ln (x))

簡素化

= x ln (ln (x))

定数を解答に追加する

= x ln (ln (x)) + C