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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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integral of 1/(x(sqrt(x)+1))

\int \frac{1}{x ( x + 1 )} d x

derivative y=2-x^2

d er i v a t i v e y = 2 - x^{2}

integral from-2pi to pi of pisin^2(x)

\int_{- 2 π}^{π} π sin^{2} (x) d x