integral of ln(x)(5x+3)

解

\int ln (x) (5 x + 3) d x

\frac{5}{2} x^{2} ln (x) - 5 x^{2} + \frac{5 x ^{2}}{4} + \frac{5 x ^{2}}{2} + 3 x ln (x) - 3 x + C

解答ステップ

\int ln (x) (5 x + 3) d x

部分積分を適用する

= (5 x + 3) (x ln (x) - x) - \int 5 (x ln (x) - x) d x

\int 5 (x ln (x) - x) d x = 5 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4} - \frac{x ^{2}}{2})

= (5 x + 3) (x ln (x) - x) - 5 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4} - \frac{x ^{2}}{2})

簡素化

(5 x + 3) (x ln (x) - x) - 5 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4} - \frac{x ^{2}}{2}) : \frac{5}{2} x^{2} ln (x) - 5 x^{2} + \frac{5 x ^{2}}{4} + \frac{5 x ^{2}}{2} + 3 x ln (x) - 3 x

= \frac{5}{2} x^{2} ln (x) - 5 x^{2} + \frac{5 x ^{2}}{4} + \frac{5 x ^{2}}{2} + 3 x ln (x) - 3 x

定数を解答に追加する

= \frac{5}{2} x^{2} ln (x) - 5 x^{2} + \frac{5 x ^{2}}{4} + \frac{5 x ^{2}}{2} + 3 x ln (x) - 3 x + C