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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

derivative of 2x+8

\frac{d}{d x} (2 x + 8)

derivative of [arctan(3x]^2)

\frac{d}{d x} ([arctan (3 x)]^{2})

derivative 1680(2x+5)^4

d er i v a t i v e 1680 (2 x + 5)^{4}

y^{''}+y=5e^{-8x}

y^{^{''}} + y = 5 e^{- 8 x}

sum from n=1 to infinity of 2(2/3)^{n-1}

n = 1 \sum \infty 2 (\frac{2}{3})^{n - 1}