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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of x^2e^{x^3-5}

\int x^{2} e^{x^{3} - 5} d x

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

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

integral of (1-sin^3(x))

\int (1 - sin^{3} (x)) d x

integral of e^{4-x}

\int e^{4 - x} d x

integral of (cos(sqrt(y+1)))/(sqrt(y+1))

\int \frac{cos ( y + 1 )}{y + 1} d y