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Beliebt Rechnen >

integral of ((x^2+x-10))/((2x-3)(x^2+4))

Lösung

\int \frac{( x ^{2} + x - 10 )}{( 2 x - 3 ) ( x ^{2} + 4 )} d x

Lösung

- \frac{1}{2} ln ∣ 2 x - 3 ∣ + \frac{1}{2} ln \frac{x ^{2}}{4} + 1 + arctan (\frac{x}{2}) + C

Schritte zur Lösung

\int \frac{x ^{2} + x - 10}{( 2 x - 3 ) ( x ^{2} + 4 )} d x

Ermittle den Partialbruch von

\frac{x ^{2} + x - 10}{( 2 x - 3 ) ( x ^{2} + 4 )} : - \frac{1}{2 x - 3} + \frac{x + 2}{x ^{2} + 4}

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

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

\int \frac{x + 2}{x ^{2} + 4} d x = \frac{1}{2} ln \frac{x ^{2}}{4} + 1 + arctan (\frac{x}{2})

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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