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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 5e^x+7

\int 5 e^{x} + 7 d x

integral of (20x)/(\sqrt[3]{7-3x^2)}

\int \frac{20 x}{3 7 - 3 x ^{2}} d x

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

\int \frac{e ^{x}}{( 1 - 3 e ^{x} ) ^{2}} d x

integral of 3(x)^{3/2}

\int 3 (x)^{\frac{3}{2}} d x

integral of (4x)/(x^2-a^2)

\int \frac{4 x}{x ^{2} - a ^{2}} d x