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Beliebt Rechnen >

integral of ((8x^2+3x-6))/((x^2-4)(x+2))

Lösung

\int \frac{( 8 x ^{2} + 3 x - 6 )}{( x ^{2} - 4 ) ( x + 2 )} d x

Lösung

6 ln ∣ x + 2 ∣ + \frac{5}{x + 2} + 2 ln ∣ x - 2 ∣ + C

Schritte zur Lösung

\int \frac{8 x ^{2} + 3 x - 6}{( x ^{2} - 4 ) ( x + 2 )} d x

Ermittle den Partialbruch von

\frac{8 x ^{2} + 3 x - 6}{( x ^{2} - 4 ) ( x + 2 )} : \frac{6}{x + 2} - \frac{5}{( x + 2 ) ^{2}} + \frac{2}{x - 2}

= \int \frac{6}{x + 2} - \frac{5}{( x + 2 ) ^{2}} + \frac{2}{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{6}{x + 2} d x - \int \frac{5}{( x + 2 ) ^{2}} d x + \int \frac{2}{x - 2} d x

\int \frac{6}{x + 2} d x = 6 ln ∣ x + 2 ∣

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

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

= 6 ln ∣ x + 2 ∣ - (- \frac{5}{x + 2}) + 2 ln ∣ x - 2 ∣

Vereinfache

= 6 ln ∣ x + 2 ∣ + \frac{5}{x + 2} + 2 ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= 6 ln ∣ x + 2 ∣ + \frac{5}{x + 2} + 2 ln ∣ x - 2 ∣ + C

Graph

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\int cot (10 x) d x

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\int \frac{x ^{2} + 24}{x ( x - 2 ) ^{3}} d x

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\frac{d}{d x} (\frac{e ^{x}}{7 + e ^{x}})