integral of (ln(x))/(1+x^2)

解

\int \frac{ln ( x )}{1 + x ^{2}} d x

ln (x) arctan (x) - \frac{1}{2} i (Li_{2} (- i x) - Li_{2} (i x)) + C

解答ステップ

\int \frac{ln ( x )}{1 + x ^{2}} d x

部分積分を適用する

= ln (x) arctan (x) - \int \frac{arctan ( x )}{x} d x

\int \frac{arctan ( x )}{x} d x = \frac{1}{2} i (Li_{2} (- i x) - Li_{2} (i x))

= ln (x) arctan (x) - \frac{1}{2} i (Li_{2} (- i x) - Li_{2} (i x))

定数を解答に追加する

= ln (x) arctan (x) - \frac{1}{2} i (Li_{2} (- i x) - Li_{2} (i x)) + C