integral of (ln(x+4))/(x^2)

解

\int \frac{ln ( x + 4 )}{x ^{2}} d x

- \frac{ln ( x + 4 )}{x} + \frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x + 4 ∣ + C

解答ステップ

\int \frac{ln ( x + 4 )}{x ^{2}} d x

部分積分を適用する

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

\int - \frac{1}{x ( x + 4 )} d x = - \frac{1}{4} ln ∣ x ∣ + \frac{1}{4} ln ∣ x + 4 ∣

= - \frac{ln ( x + 4 )}{x} - (- \frac{1}{4} ln ∣ x ∣ + \frac{1}{4} ln ∣ x + 4 ∣)

簡素化

= - \frac{ln ( x + 4 )}{x} + \frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x + 4 ∣

定数を解答に追加する

= - \frac{ln ( x + 4 )}{x} + \frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x + 4 ∣ + C