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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Faktorisiere

x^{2} - 1 : - (- x^{2} + 1)

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

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

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

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

\int 1 d x = x

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

derivative of (1+x^2^2)

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

implicit (d^2y)/(dx^2),xy+7ey(x)=7e

im pl i c i t \frac{d ^{2} y}{d x ^{2}}, x y + 7 ey (x) = 7 e

derivative of (2xcos(x^2)/(sin(x^2)+1))

\frac{d}{d x} (\frac{2 x cos ( x ^{2} )}{sin ( x ^{2} ) + 1})

integral of x/(x^2-16)

\int \frac{x}{x ^{2} - 16} d x

(x+y)(dy)/(dx)=1

(x + y) \frac{d y}{d x} = 1