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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{7}{25} ln ∣ x + 3 ∣ + \frac{7}{25} ln ∣ x - 2 ∣ - \frac{8}{5 ( x + 3 )} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Vereinfache

3 (- \frac{2}{25} ln ∣ x + 3 ∣ - \frac{3}{5 ( x + 3 )} + \frac{2}{25} ln ∣ x - 2 ∣) + \frac{1}{25} ln ∣ x - 2 ∣ - \frac{1}{25} ln ∣ x + 3 ∣ + \frac{1}{5 ( x + 3 )} : - \frac{7}{25} ln ∣ x + 3 ∣ + \frac{7}{25} ln ∣ x - 2 ∣ - \frac{8}{5 ( x + 3 )}

= - \frac{7}{25} ln ∣ x + 3 ∣ + \frac{7}{25} ln ∣ x - 2 ∣ - \frac{8}{5 ( x + 3 )}

Füge eine Konstante zur Lösung hinzu

= - \frac{7}{25} ln ∣ x + 3 ∣ + \frac{7}{25} ln ∣ x - 2 ∣ - \frac{8}{5 ( x + 3 )} + C

Graph

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