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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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