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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{2}{x + 1} d x = 2 ln ∣ x + 1 ∣

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 1/(sqrt(x^2+y^2))

\int \frac{1}{x ^{2} + y ^{2}} d x

tangent f(x)=x^4+9x^2-x,\at x=1

t an g e n t f (x) = x^{4} + 9 x^{2} - x, at x = 1

sum from n=3 to infinity of (4/7)^n

n = 3 \sum \infty (\frac{4}{7})^{n}

integral of 1/(cos^2(4x))

\int \frac{1}{cos ^{2} ( 4 x )} d x

derivative of e^x-4

\frac{d}{d x} (e^{x} - 4)