integral of (sqrt(x^2+5))/(x^2)

解

\int \frac{x ^{2} + 5}{x ^{2}} d x

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

解答ステップ

\int \frac{x ^{2} + 5}{x ^{2}} d x

部分積分を適用する

= - \frac{x ^{2} + 5}{x} - \int - \frac{1}{x ^{2} + 5} d x

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

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

簡素化

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

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

定数を解答に追加する

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