integral of (x^5+4)ln(x)

解

\int (x^{5} + 4) ln (x) d x

\frac{1}{6} x^{6} ln (x) - \frac{x ^{6}}{36} + 4 (x ln (x) - x) + C

解答ステップ

\int (x^{5} + 4) ln (x) d x

拡張

(x^{5} + 4) ln (x) : x^{5} ln (x) + 4 ln (x)

= \int x^{5} ln (x) + 4 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 x^{5} ln (x) d x + \int 4 ln (x) d x

\int x^{5} ln (x) d x = \frac{1}{6} x^{6} ln (x) - \frac{x ^{6}}{36}

\int 4 ln (x) d x = 4 (x ln (x) - x)

= \frac{1}{6} x^{6} ln (x) - \frac{x ^{6}}{36} + 4 (x ln (x) - x)

定数を解答に追加する

= \frac{1}{6} x^{6} ln (x) - \frac{x ^{6}}{36} + 4 (x ln (x) - x) + C