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Beliebt Rechnen >

integral of (x^2+x-5)/(x^2-1)

Lösung

\int \frac{x ^{2} + x - 5}{x ^{2} - 1} d x

Lösung

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

Schritte zur Lösung

\int \frac{x ^{2} + x - 5}{x ^{2} - 1} d x

Multipliziere aus

\frac{x ^{2} + x - 5}{x ^{2} - 1} : \frac{x ^{2}}{x ^{2} - 1} + \frac{x}{x ^{2} - 1} - \frac{5}{x ^{2} - 1}

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 ^{2} - 1} d x + \int \frac{x}{x ^{2} - 1} d x - \int \frac{5}{x ^{2} - 1} d x

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

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

\int \frac{5}{x ^{2} - 1} d x = - 5 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣)

= - \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} ln ∣ x - 1 ∣ + x + \frac{1}{2} ln x^{2} - 1 - (- 5 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣))

Vereinfache

- \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} ln ∣ x - 1 ∣ + x + \frac{1}{2} ln x^{2} - 1 - (- 5 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣)) : x + 2 ln ∣ x + 1 ∣ - 2 ln ∣ x - 1 ∣ + \frac{1}{2} ln x^{2} - 1

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 5/(8x(x+1)^2)

\int \frac{5}{8 x ( x + 1 ) ^{2}} d x

integral of-e^{-2y}+1

\int - e^{- 2 y} + 1 d y

tangent y= x/(x-5),(-1,a)

t an g e n t y = \frac{x}{x - 5}, (- 1, a)

integral from 6 to 8 of (48)/((x-6)^3)

\int_{6}^{8} \frac{48}{( x - 6 ) ^{3}} d x

(\partial)/(\partial x)(x/(y+8z))

\frac{\partial}{\partial x} (\frac{x}{y + 8 z})