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

解

\int (ln (x) + x ln (x)) d x

x ln (x) - x + \frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4} + C

解答ステップ

\int ln (x) + 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 ln (x) d x + \int x ln (x) d x

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

\int x ln (x) d x = \frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}

= x ln (x) - x + \frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}

定数を解答に追加する

= x ln (x) - x + \frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4} + C