integral of (x^3+2x^2-3x+1)/(x^3-27)

해법

\int \frac{x ^{3} + 2 x ^{2} - 3 x + 1}{x ^{3} - 27} d x

x - \frac{14}{27} ln 4 x^{2} + 12 x + 36 + \frac{2}{3} ln x^{3} - 27 + \frac{19}{27} ln ∣ x - 3 ∣ - \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3}) - 3 arctan (\frac{2}{3 3} x + \frac{1}{3}) - \frac{1}{6} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{2}{3 3} x + \frac{1}{3})) + C

솔루션 단계

\int \frac{x ^{3} + 2 x ^{2} - 3 x + 1}{x ^{3} - 27} d x

\frac{x ^{3} + 2 x ^{2} - 3 x + 1}{x ^{3} - 27}

확대한다:

\frac{x ^{3}}{x ^{3} - 27} + \frac{2 x ^{2}}{x ^{3} - 27} - \frac{3 x}{x ^{3} - 27} + \frac{1}{x ^{3} - 27}

합계 규칙 적용:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{3}}{x ^{3} - 27} d x + \int \frac{2 x ^{2}}{x ^{3} - 27} d x - \int \frac{3 x}{x ^{3} - 27} d x + \int \frac{1}{x ^{3} - 27} d x

\int \frac{x ^{3}}{x ^{3} - 27} d x = x + ln ∣ x - 3 ∣ - \frac{1}{2} ln 4 x^{2} + 12 x + 36 - 3 arctan (\frac{2}{3 3} x + \frac{1}{3})

\int \frac{2 x ^{2}}{x ^{3} - 27} d x = \frac{2}{3} ln x^{3} - 27

\int \frac{3 x}{x ^{3} - 27} d x = 3 (\frac{1}{9} ln ∣ x - 3 ∣ + \frac{1}{18} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{1}{3 3} (2 x + 3))))

\int \frac{1}{x ^{3} - 27} d x = \frac{1}{27} ln ∣ x - 3 ∣ - \frac{1}{54} ln 4 x^{2} + 12 x + 36 - \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3})

= x + ln ∣ x - 3 ∣ - \frac{1}{2} ln 4 x^{2} + 12 x + 36 - 3 arctan (\frac{2}{3 3} x + \frac{1}{3}) + \frac{2}{3} ln x^{3} - 27 - 3 (\frac{1}{9} ln ∣ x - 3 ∣ + \frac{1}{18} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{1}{3 3} (2 x + 3)))) + \frac{1}{27} ln ∣ x - 3 ∣ - \frac{1}{54} ln 4 x^{2} + 12 x + 36 - \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3})

x + ln ∣ x - 3 ∣ - \frac{1}{2} ln 4 x^{2} + 12 x + 36 - 3 arctan (\frac{2}{3 3} x + \frac{1}{3}) + \frac{2}{3} ln x^{3} - 27 - 3 (\frac{1}{9} ln ∣ x - 3 ∣ + \frac{1}{18} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{1}{3 3} (2 x + 3)))) + \frac{1}{27} ln ∣ x - 3 ∣ - \frac{1}{54} ln 4 x^{2} + 12 x + 36 - \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3})

간소화하다 :

x - \frac{14}{27} ln 4 x^{2} + 12 x + 36 + \frac{2}{3} ln x^{3} - 27 + \frac{19}{27} ln ∣ x - 3 ∣ - \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3}) - 3 arctan (\frac{2}{3 3} x + \frac{1}{3}) - \frac{1}{6} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{2}{3 3} x + \frac{1}{3}))

= x - \frac{14}{27} ln 4 x^{2} + 12 x + 36 + \frac{2}{3} ln x^{3} - 27 + \frac{19}{27} ln ∣ x - 3 ∣ - \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3}) - 3 arctan (\frac{2}{3 3} x + \frac{1}{3}) - \frac{1}{6} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{2}{3 3} x + \frac{1}{3}))

솔루션에 상수 추가

= x - \frac{14}{27} ln 4 x^{2} + 12 x + 36 + \frac{2}{3} ln x^{3} - 27 + \frac{19}{27} ln ∣ x - 3 ∣ - \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3}) - 3 arctan (\frac{2}{3 3} x + \frac{1}{3}) - \frac{1}{6} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{2}{3 3} x + \frac{1}{3})) + C