integral of (25+x^2)/(5x)

解

\int \frac{25 + x ^{2}}{5 x} d x

\frac{1}{5} (25 ln ∣ x ∣ + \frac{x ^{2}}{2}) + C

解答ステップ

\int \frac{25 + x ^{2}}{5 x} d x

定数を除く:

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

= \frac{1}{5} \cdot \int \frac{25 + x ^{2}}{x} d x

拡張

\frac{25 + x ^{2}}{x} : \frac{25}{x} + x

= \frac{1}{5} \cdot \int \frac{25}{x} + x 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

= \frac{1}{5} (\int \frac{25}{x} d x + \int x d x)

\int \frac{25}{x} d x = 25 ln ∣ x ∣

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

= \frac{1}{5} (25 ln ∣ x ∣ + \frac{x ^{2}}{2})

定数を解答に追加する

= \frac{1}{5} (25 ln ∣ x ∣ + \frac{x ^{2}}{2}) + C