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Beliebt Rechnen >

integral of (-4x+8)/((x^2+1)(x-1))

Lösung

\int \frac{- 4 x + 8}{( x ^{2} + 1 ) ( x - 1 )} d x

Lösung

- ln x^{2} + 1 - 6 arctan (x) + 2 ln ∣ x - 1 ∣ + C

Schritte zur Lösung

\int \frac{- 4 x + 8}{( x ^{2} + 1 ) ( x - 1 )} d x

Multipliziere aus

\frac{- 4 x + 8}{( x ^{2} + 1 ) ( x - 1 )} : - \frac{4 x}{( x ^{2} + 1 ) ( x - 1 )} + \frac{8}{( x ^{2} + 1 ) ( x - 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{4 x}{( x ^{2} + 1 ) ( x - 1 )} d x + \int \frac{8}{( x ^{2} + 1 ) ( x - 1 )} d x

\int \frac{4 x}{( x ^{2} + 1 ) ( x - 1 )} d x = 4 (\frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 + arctan (x)) + \frac{1}{2} ln ∣ x - 1 ∣)

\int \frac{8}{( x ^{2} + 1 ) ( x - 1 )} d x = 8 (\frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 - arctan (x)) + \frac{1}{2} ln ∣ x - 1 ∣)

= - 4 (\frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 + arctan (x)) + \frac{1}{2} ln ∣ x - 1 ∣) + 8 (\frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 - arctan (x)) + \frac{1}{2} ln ∣ x - 1 ∣)

Vereinfache

- 4 (\frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 + arctan (x)) + \frac{1}{2} ln ∣ x - 1 ∣) + 8 (\frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 - arctan (x)) + \frac{1}{2} ln ∣ x - 1 ∣) : - ln x^{2} + 1 - 6 arctan (x) + 2 ln ∣ x - 1 ∣

= - ln x^{2} + 1 - 6 arctan (x) + 2 ln ∣ x - 1 ∣

Füge eine Konstante zur Lösung hinzu

= - ln x^{2} + 1 - 6 arctan (x) + 2 ln ∣ x - 1 ∣ + C

Graph

Beliebte Beispiele

integral of sqrt(x^2-6)

\int x^{2} - 6 d x

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

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

integral of (x^2+7x-6)/((x+1)(x^2-4x+4))

\int \frac{x ^{2} + 7 x - 6}{( x + 1 ) ( x ^{2} - 4 x + 4 )} d x

integral of \sqrt[4]{y}

\int 4 y d y

integral of x/((x-6))

\int \frac{x}{( x - 6 )} d x