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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (x-8)/(x^2-x-6)

\int \frac{x - 8}{x ^{2} - x - 6} d x

integral of ((x^2)/3-8x-3)

\int (\frac{x ^{2}}{3} - 8 x - 3) d x

integral of sin^5(x)cos^9(x)

\int sin^{5} (x) cos^{9} (x) d x

integral of (x+1)/(x^2+10x+18)

\int \frac{x + 1}{x ^{2} + 10 x + 18} d x

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

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