integral of xln(sqrt(5)x)

解

\int x ln (5 x) d x

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

解答ステップ

\int x ln (5 x) d x

拡張

x ln (5 x) : x ln (x) + \frac{ln ( 5 ) x}{2}

= \int x ln (x) + \frac{ln ( 5 ) x}{2} 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 ln (x) d x + \int \frac{ln ( 5 ) x}{2} d x

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

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

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

定数を解答に追加する

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