integral of (2x^2+24)/(x(x-2)^3)

Lösung

\int \frac{2 x ^{2} + 24}{x ( x - 2 ) ^{3}} d x

\frac{4}{x - 2} - \frac{8}{( x - 2 ) ^{2}} - 3 ln ∣ x ∣ + 3 ln ∣ x - 2 ∣ + C

Schritte zur Lösung

\int \frac{2 x ^{2} + 24}{x ( x - 2 ) ^{3}} d x

Multipliziere aus

\frac{2 x ^{2} + 24}{x ( x - 2 ) ^{3}} : \frac{2 x}{( x - 2 ) ^{3}} + \frac{24}{x ( x - 2 ) ^{3}}

= \int \frac{2 x}{( x - 2 ) ^{3}} + \frac{24}{x ( x - 2 ) ^{3}} d x

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{2 x}{( x - 2 ) ^{3}} d x + \int \frac{24}{x ( x - 2 ) ^{3}} d x

\int \frac{2 x}{( x - 2 ) ^{3}} d x = 2 (- \frac{1}{x - 2} - \frac{1}{( x - 2 ) ^{2}})

\int \frac{24}{x ( x - 2 ) ^{3}} d x = 24 (- \frac{1}{8} ln ∣ x ∣ + \frac{1}{8} ln ∣ x - 2 ∣ + \frac{1}{4 ( x - 2 )} - \frac{1}{4 ( x - 2 ) ^{2}})

= 2 (- \frac{1}{x - 2} - \frac{1}{( x - 2 ) ^{2}}) + 24 (- \frac{1}{8} ln ∣ x ∣ + \frac{1}{8} ln ∣ x - 2 ∣ + \frac{1}{4 ( x - 2 )} - \frac{1}{4 ( x - 2 ) ^{2}})

Vereinfache

2 (- \frac{1}{x - 2} - \frac{1}{( x - 2 ) ^{2}}) + 24 (- \frac{1}{8} ln ∣ x ∣ + \frac{1}{8} ln ∣ x - 2 ∣ + \frac{1}{4 ( x - 2 )} - \frac{1}{4 ( x - 2 ) ^{2}}) : \frac{4}{x - 2} - \frac{8}{( x - 2 ) ^{2}} - 3 ln ∣ x ∣ + 3 ln ∣ x - 2 ∣

= \frac{4}{x - 2} - \frac{8}{( x - 2 ) ^{2}} - 3 ln ∣ x ∣ + 3 ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{4}{x - 2} - \frac{8}{( x - 2 ) ^{2}} - 3 ln ∣ x ∣ + 3 ln ∣ x - 2 ∣ + C

\frac{d}{d x} (- x^{2} - 3 x + 4)

\int \frac{1}{a ^{2} - u ^{2}} d u

θ^{^{''}} + \frac{a}{b} \cdot θ = 0

\int_{2}^{5} \frac{1}{x} d x

t an g e n t x^{2} - 1