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

Lösung

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

- 2 ln ∣ x - 2 ∣ - \frac{1}{x - 2} + ln ∣ x ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

Wende die Summenregel an:

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

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

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

\int \frac{x}{x ( x - 2 ) ^{2}} d x = - \frac{1}{x - 2}

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

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

Vereinfache

- (ln ∣ x - 2 ∣ - \frac{2}{x - 2}) - \frac{1}{x - 2} + ln ∣ x ∣ - ln ∣ x - 2 ∣ - \frac{2}{x - 2} : - 2 ln ∣ x - 2 ∣ - \frac{1}{x - 2} + ln ∣ x ∣

= - 2 ln ∣ x - 2 ∣ - \frac{1}{x - 2} + ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= - 2 ln ∣ x - 2 ∣ - \frac{1}{x - 2} + ln ∣ x ∣ + C

n = 1 \sum \infty ln (\frac{n + 1}{n})

9 y^{^{''}} - y = \frac{x e x}{3}

\int_{3}^{8} x^{3} ln (3 x) d x

y^{^{''}} - 8 y^{^{'}} + 20 y = 200 x^{2} - 39 x e^{x}

\int_{0}^{8} x e^{- x} d x