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Beliebt Rechnen >

integral of (2x-7)/(x^2-12x+32)

Lösung

\int \frac{2 x - 7}{x ^{2} - 12 x + 32} d x

Lösung

- \frac{1}{4} ln ∣ x - 4 ∣ + \frac{9}{4} ln ∣ x - 8 ∣ + C

Schritte zur Lösung

\int \frac{2 x - 7}{x ^{2} - 12 x + 32} d x

Vervollständige das Quadrat

x^{2} - 12 x + 32 : (x - 6)^{2} - 4

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

Wende U-Substitution an

= \int \frac{2 u + 5}{u ^{2} - 4} d u

Ermittle den Partialbruch von

\frac{2 u + 5}{u ^{2} - 4} : - \frac{1}{4 ( u + 2 )} + \frac{9}{4 ( u - 2 )}

= \int - \frac{1}{4 ( u + 2 )} + \frac{9}{4 ( u - 2 )} d u

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}{4 ( u + 2 )} d u + \int \frac{9}{4 ( u - 2 )} d u

\int \frac{1}{4 ( u + 2 )} d u = \frac{1}{4} ln ∣ u + 2 ∣

\int \frac{9}{4 ( u - 2 )} d u = \frac{9}{4} ln ∣ u - 2 ∣

= - \frac{1}{4} ln ∣ u + 2 ∣ + \frac{9}{4} ln ∣ u - 2 ∣

Setze in

u = x - 6

ein

= - \frac{1}{4} ln ∣ x - 6 + 2 ∣ + \frac{9}{4} ln ∣ x - 6 - 2 ∣

Vereinfache

- \frac{1}{4} ln ∣ x - 6 + 2 ∣ + \frac{9}{4} ln ∣ x - 6 - 2 ∣ : - \frac{1}{4} ln ∣ x - 4 ∣ + \frac{9}{4} ln ∣ x - 8 ∣

= - \frac{1}{4} ln ∣ x - 4 ∣ + \frac{9}{4} ln ∣ x - 8 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{4} ln ∣ x - 4 ∣ + \frac{9}{4} ln ∣ x - 8 ∣ + C

Graph

Beliebte Beispiele

integral of ((4x^3))/(x^4+1)

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

integral of (xy^2+3x^2y)

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

integral of (2x-7)/((x+1)(x-3))

\int \frac{2 x - 7}{( x + 1 ) ( x - 3 )} d x

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

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

integral of 4/(xsqrt(x))

\int \frac{4}{x x} d x