integral of (8x-3)ln(x)

解

\int (8 x - 3) ln (x) d x

8 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}) - 3 (x ln (x) - x) + C

解答ステップ

\int (8 x - 3) ln (x) d x

拡張

(8 x - 3) ln (x) : 8 x ln (x) - 3 ln (x)

= \int 8 x ln (x) - 3 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 8 x ln (x) d x - \int 3 ln (x) d x

\int 8 x ln (x) d x = 8 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4})

\int 3 ln (x) d x = 3 (x ln (x) - x)

= 8 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}) - 3 (x ln (x) - x)

定数を解答に追加する

= 8 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}) - 3 (x ln (x) - x) + C