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

Lösung

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

\frac{1}{x - 1} - \frac{1}{( x - 1 ) ^{2}} - 3 ln ∣ x ∣ + 3 ln ∣ x - 1 ∣ - \frac{3}{2 ( x - 1 ) ^{2}} + C

Schritte zur Lösung

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

Multipliziere aus

\frac{2 x ^{2} + 3}{x ( x - 1 ) ^{3}} : \frac{2 x}{( x - 1 ) ^{3}} + \frac{3}{x ( x - 1 ) ^{3}}

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

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

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

= - \frac{2}{x - 1} - \frac{1}{( x - 1 ) ^{2}} + 3 (- ln ∣ x ∣ + ln ∣ x - 1 ∣ + \frac{1}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}})

Vereinfache

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

= \frac{1}{x - 1} - \frac{1}{( x - 1 ) ^{2}} - 3 ln ∣ x ∣ + 3 ln ∣ x - 1 ∣ - \frac{3}{2 ( x - 1 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{x - 1} - \frac{1}{( x - 1 ) ^{2}} - 3 ln ∣ x ∣ + 3 ln ∣ x - 1 ∣ - \frac{3}{2 ( x - 1 ) ^{2}} + C

\frac{d y}{d x} = \frac{2 y}{x + 1}

x \to 1 lim (x + 4)

n = 1 \sum \infty \frac{n}{( 5 ^{n} )}

\frac{d}{d x} ((4 x^{2} + 1)^{- 50})

\int (\frac{x ^{2} + 3 x - 5}{x - 3}) d x