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Beliebt Rechnen >

integral of (3x^2-4x+5)/((x-1)(x^2+1))

Lösung

\int \frac{3 x ^{2} - 4 x + 5}{( x - 1 ) ( x ^{2} + 1 )} d x

Lösung

2 ln ∣ x - 1 ∣ + \frac{1}{2} ln x^{2} + 1 - 3 arctan (x) + C

Schritte zur Lösung

\int \frac{3 x ^{2} - 4 x + 5}{( x - 1 ) ( x ^{2} + 1 )} d x

Ermittle den Partialbruch von

\frac{3 x ^{2} - 4 x + 5}{( x - 1 ) ( x ^{2} + 1 )} : \frac{2}{x - 1} + \frac{x - 3}{x ^{2} + 1}

= \int \frac{2}{x - 1} + \frac{x - 3}{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{2}{x - 1} d x + \int \frac{x - 3}{x ^{2} + 1} d x

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

\int \frac{x - 3}{x ^{2} + 1} d x = \frac{1}{2} ln x^{2} + 1 - 3 arctan (x)

= 2 ln ∣ x - 1 ∣ + \frac{1}{2} ln x^{2} + 1 - 3 arctan (x)

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ x - 1 ∣ + \frac{1}{2} ln x^{2} + 1 - 3 arctan (x) + C

Graph

Beliebte Beispiele

derivative of (sin(2x)^x)

\frac{d}{d x} ((sin (2 x))^{x})

integral of 3(5^{1-2x})

\int 3 (5^{1 - 2 x}) d x

derivative of 40sqrt(x)

\frac{d}{d x} (40 x)

limit as x approaches-2 of 3x-1

x \to - 2 lim (3 x - 1)

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

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