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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{5}{2} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣ + \frac{13}{4} ln ∣ x + 3 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{2} - 4 x + 5}{( x ^{2} - 1 ) ( x + 3 )} : - \frac{5}{2 ( x + 1 )} + \frac{1}{4 ( x - 1 )} + \frac{13}{4 ( x + 3 )}

= \int - \frac{5}{2 ( x + 1 )} + \frac{1}{4 ( x - 1 )} + \frac{13}{4 ( x + 3 )} 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{5}{2 ( x + 1 )} d x + \int \frac{1}{4 ( x - 1 )} d x + \int \frac{13}{4 ( x + 3 )} d x

\int \frac{5}{2 ( x + 1 )} d x = \frac{5}{2} ln ∣ x + 1 ∣

\int \frac{1}{4 ( x - 1 )} d x = \frac{1}{4} ln ∣ x - 1 ∣

\int \frac{13}{4 ( x + 3 )} d x = \frac{13}{4} ln ∣ x + 3 ∣

= - \frac{5}{2} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣ + \frac{13}{4} ln ∣ x + 3 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{5}{2} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣ + \frac{13}{4} ln ∣ x + 3 ∣ + C

Graph

Beliebte Beispiele

integral from 1 to 4 of xln(x)

\int_{1}^{4} x ln (x) d x

tangent f(x)=2x^2-9x+10,\at x=1

t an g e n t f (x) = 2 x^{2} - 9 x + 10, at x = 1

y^'+1/x y= 2/3 x^4y^4

y^{^{'}} + \frac{1}{x} y = \frac{2}{3} x^{4} y^{4}

derivative of 7e^xcsc(x)

\frac{d}{d x} (7 e^{x} csc (x))

derivative (1-3x+x^3)/(x^3)

d er i v a t i v e \frac{1 - 3 x + x ^{3}}{x ^{3}}