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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{2}{25} ln ∣ x + 3 ∣ - \frac{2}{25} ln ∣ x - 2 ∣ - \frac{2}{x - 2} + \frac{3}{5 ( x - 2 )} + C

Schritte zur Lösung

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

Multipliziere aus

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

\int \frac{x}{( x + 3 ) ( x - 2 ) ^{2}} d x = - \frac{1}{x - 2} - 3 (\frac{1}{25} ln ∣ x + 3 ∣ - \frac{1}{25} ln ∣ x - 2 ∣ - \frac{1}{5 ( x - 2 )})

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

= - \frac{1}{x - 2} - 3 (\frac{1}{25} ln ∣ x + 3 ∣ - \frac{1}{25} ln ∣ x - 2 ∣ - \frac{1}{5 ( x - 2 )}) + \frac{1}{5} ln ∣ x + 3 ∣ - \frac{1}{5} ln ∣ x - 2 ∣ - \frac{1}{x - 2}

Vereinfache

- \frac{1}{x - 2} - 3 (\frac{1}{25} ln ∣ x + 3 ∣ - \frac{1}{25} ln ∣ x - 2 ∣ - \frac{1}{5 ( x - 2 )}) + \frac{1}{5} ln ∣ x + 3 ∣ - \frac{1}{5} ln ∣ x - 2 ∣ - \frac{1}{x - 2} : \frac{2}{25} ln ∣ x + 3 ∣ - \frac{2}{25} ln ∣ x - 2 ∣ - \frac{2}{x - 2} + \frac{3}{5 ( x - 2 )}

= \frac{2}{25} ln ∣ x + 3 ∣ - \frac{2}{25} ln ∣ x - 2 ∣ - \frac{2}{x - 2} + \frac{3}{5 ( x - 2 )}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{25} ln ∣ x + 3 ∣ - \frac{2}{25} ln ∣ x - 2 ∣ - \frac{2}{x - 2} + \frac{3}{5 ( x - 2 )} + C

Graph

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