integral of (x+8)/(x^3+48x-49)

解

\int \frac{x + 8}{x ^{3} + 48 x - 49} d x

\frac{3}{17} ln ∣ x - 1 ∣ - \frac{3}{34} ln 4 x^{2} + 4 x + 196 + \frac{5 5}{17 39} arctan (\frac{1}{195} (2 x + 1)) + C

解答ステップ

\int \frac{x + 8}{x ^{3} + 48 x - 49} d x

拡張

\frac{x + 8}{x ^{3} + 48 x - 49} : \frac{x}{x ^{3} + 48 x - 49} + \frac{8}{x ^{3} + 48 x - 49}

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 ^{3} + 48 x - 49} d x + \int \frac{8}{x ^{3} + 48 x - 49} d x

\int \frac{x}{x ^{3} + 48 x - 49} d x = \frac{1}{51} ln ∣ x - 1 ∣ - \frac{1}{102} ln 4 x^{2} + 4 x + 196 + \frac{11 3}{17 65} arctan (\frac{1}{195} (2 x + 1))

\int \frac{8}{x ^{3} + 48 x - 49} d x = 8 (\frac{1}{51} ln ∣ x - 1 ∣ - \frac{1}{102} ln 4 x^{2} + 4 x + 196 - \frac{1}{17 195} arctan (\frac{1}{195} (2 x + 1)))

= \frac{1}{51} ln ∣ x - 1 ∣ - \frac{1}{102} ln 4 x^{2} + 4 x + 196 + \frac{11 3}{17 65} arctan (\frac{1}{195} (2 x + 1)) + 8 (\frac{1}{51} ln ∣ x - 1 ∣ - \frac{1}{102} ln 4 x^{2} + 4 x + 196 - \frac{1}{17 195} arctan (\frac{1}{195} (2 x + 1)))

簡素化

\frac{1}{51} ln ∣ x - 1 ∣ - \frac{1}{102} ln 4 x^{2} + 4 x + 196 + \frac{11 3}{17 65} arctan (\frac{1}{195} (2 x + 1)) + 8 (\frac{1}{51} ln ∣ x - 1 ∣ - \frac{1}{102} ln 4 x^{2} + 4 x + 196 - \frac{1}{17 195} arctan (\frac{1}{195} (2 x + 1))) : \frac{3}{17} ln ∣ x - 1 ∣ - \frac{3}{34} ln 4 x^{2} + 4 x + 196 + \frac{5 5}{17 39} arctan (\frac{1}{195} (2 x + 1))

= \frac{3}{17} ln ∣ x - 1 ∣ - \frac{3}{34} ln 4 x^{2} + 4 x + 196 + \frac{5 5}{17 39} arctan (\frac{1}{195} (2 x + 1))

定数を解答に追加する

= \frac{3}{17} ln ∣ x - 1 ∣ - \frac{3}{34} ln 4 x^{2} + 4 x + 196 + \frac{5 5}{17 39} arctan (\frac{1}{195} (2 x + 1)) + C