integral of (8ln(x+9))/(x^2)

解

\int \frac{8 ln ( x + 9 )}{x ^{2}} d x

8 (- \frac{ln ( x + 9 )}{x} + \frac{1}{9} ln ∣ x ∣ - \frac{1}{9} ln ∣ x + 9 ∣) + C

解答ステップ

\int \frac{8 ln ( x + 9 )}{x ^{2}} d x

定数を除く:

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

= 8 \cdot \int \frac{ln ( x + 9 )}{x ^{2}} d x

部分積分を適用する

= 8 (- \frac{ln ( x + 9 )}{x} - \int - \frac{1}{x ( x + 9 )} d x)

\int - \frac{1}{x ( x + 9 )} d x = - \frac{1}{9} ln ∣ x ∣ + \frac{1}{9} ln ∣ x + 9 ∣

= 8 (- \frac{ln ( x + 9 )}{x} - (- \frac{1}{9} ln ∣ x ∣ + \frac{1}{9} ln ∣ x + 9 ∣))

簡素化

= 8 (- \frac{ln ( x + 9 )}{x} + \frac{1}{9} ln ∣ x ∣ - \frac{1}{9} ln ∣ x + 9 ∣)

定数を解答に追加する

= 8 (- \frac{ln ( x + 9 )}{x} + \frac{1}{9} ln ∣ x ∣ - \frac{1}{9} ln ∣ x + 9 ∣) + C