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Beliebt Rechnen >

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

Lösung

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

Lösung

3 x + ln ∣ x - 1 ∣ - 4 ln ∣ x + 2 ∣ + C

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Ermittle den Partialbruch von

\frac{x ^{2}}{x ^{2} + x - 2} : 1 + \frac{1}{3 ( x - 1 )} - \frac{4}{3 ( x + 2 )}

= 3 \cdot \int 1 + \frac{1}{3 ( x - 1 )} - \frac{4}{3 ( 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

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

\int 1 d x = x

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

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

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

Vereinfache

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

= 3 x + ln ∣ x - 1 ∣ - 4 ln ∣ x + 2 ∣

Füge eine Konstante zur Lösung hinzu

= 3 x + ln ∣ x - 1 ∣ - 4 ln ∣ x + 2 ∣ + C

Graph

Beliebte Beispiele

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

\int (x - 2) (4 x + 2) d x

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\int 4.4 e^{(- 3 x + 4)} d x

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

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\int \frac{8 x}{1 + 4 x ^{2}} d x

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\int \frac{5}{x ( x ^{2} + 4 ) ^{2}} d x