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Beliebt Rechnen >

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

Lösung

\int \frac{x - 4}{x ^{2} - 3 x + 2} d x

Lösung

\frac{1}{2} ln 4 x^{2} - 12 x + 8 + \frac{5}{2} ln ∣ 2 x - 2 ∣ - \frac{5}{2} ln ∣ 2 x - 4 ∣ + C

Schritte zur Lösung

\int \frac{x - 4}{x ^{2} - 3 x + 2} d x

Multipliziere aus

\frac{x - 4}{x ^{2} - 3 x + 2} : \frac{x}{x ^{2} - 3 x + 2} - \frac{4}{x ^{2} - 3 x + 2}

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{x}{x ^{2} - 3 x + 2} d x - \int \frac{4}{x ^{2} - 3 x + 2} d x

\int \frac{x}{x ^{2} - 3 x + 2} d x = 2 (\frac{1}{4} ln 4 x^{2} - 12 x + 8 - \frac{3}{4} (ln ∣ 2 x - 2 ∣ - ln ∣ 2 x - 4 ∣))

\int \frac{4}{x ^{2} - 3 x + 2} d x = - 8 (\frac{1}{2} ln ∣ 2 x - 2 ∣ - \frac{1}{2} ln ∣ 2 x - 4 ∣)

= 2 (\frac{1}{4} ln 4 x^{2} - 12 x + 8 - \frac{3}{4} (ln ∣ 2 x - 2 ∣ - ln ∣ 2 x - 4 ∣)) - (- 8 (\frac{1}{2} ln ∣ 2 x - 2 ∣ - \frac{1}{2} ln ∣ 2 x - 4 ∣))

Vereinfache

2 (\frac{1}{4} ln 4 x^{2} - 12 x + 8 - \frac{3}{4} (ln ∣ 2 x - 2 ∣ - ln ∣ 2 x - 4 ∣)) - (- 8 (\frac{1}{2} ln ∣ 2 x - 2 ∣ - \frac{1}{2} ln ∣ 2 x - 4 ∣)) : \frac{1}{2} ln 4 x^{2} - 12 x + 8 + \frac{5}{2} ln ∣ 2 x - 2 ∣ - \frac{5}{2} ln ∣ 2 x - 4 ∣

= \frac{1}{2} ln 4 x^{2} - 12 x + 8 + \frac{5}{2} ln ∣ 2 x - 2 ∣ - \frac{5}{2} ln ∣ 2 x - 4 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln 4 x^{2} - 12 x + 8 + \frac{5}{2} ln ∣ 2 x - 2 ∣ - \frac{5}{2} ln ∣ 2 x - 4 ∣ + C

Graph

Beliebte Beispiele

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

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

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\int \frac{a ^{3}}{x ^{2} + a ^{2}} d x

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

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

integral of sin(18x)cos(15x)

\int sin (18 x) cos (15 x) d x

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\int \frac{1}{3 x - 9} d x