integral of (ln(x^2))/(sqrt(x))

解

\int \frac{ln ( x ^{2} )}{x} d x

2 (2 x ln (x) - 4 x) + C

解答ステップ

\int \frac{ln ( x ^{2} )}{x} d x

簡素化

ln (x^{2}) : 2 ln (x)

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

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

部分積分を適用する

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

\int \frac{2}{x} d x = 4 x

= 2 (2 x ln (x) - 4 x)

定数を解答に追加する

= 2 (2 x ln (x) - 4 x) + C