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

解

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

\frac{1}{2} arctan (\frac{x}{2}) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 4 + C

解答ステップ

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

拡張

\frac{x + 4}{x ( x ^{2} + 4 )} : \frac{x}{x ( x ^{2} + 4 )} + \frac{4}{x ( x ^{2} + 4 )}

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

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

\int \frac{x}{x ( x ^{2} + 4 )} d x = \frac{1}{2} arctan (\frac{x}{2})

\int \frac{4}{x ( x ^{2} + 4 )} d x = ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 4

= \frac{1}{2} arctan (\frac{x}{2}) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 4

定数を解答に追加する

= \frac{1}{2} arctan (\frac{x}{2}) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 4 + C