integral of (5x+12)/(x(x^2+4))

解

\int \frac{5 x + 12}{x ( x ^{2} + 4 )} d x

\frac{5}{2} arctan (\frac{x}{2}) + 12 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4) + C

解答ステップ

\int \frac{5 x + 12}{x ( x ^{2} + 4 )} d x

拡張

\frac{5 x + 12}{x ( x ^{2} + 4 )} : \frac{5 x}{x ( x ^{2} + 4 )} + \frac{12}{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{5 x}{x ( x ^{2} + 4 )} d x + \int \frac{12}{x ( x ^{2} + 4 )} d x

\int \frac{5 x}{x ( x ^{2} + 4 )} d x = \frac{5}{2} arctan (\frac{x}{2})

\int \frac{12}{x ( x ^{2} + 4 )} d x = 12 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4)

= \frac{5}{2} arctan (\frac{x}{2}) + 12 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4)

定数を解答に追加する

= \frac{5}{2} arctan (\frac{x}{2}) + 12 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4) + C