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Beliebt Rechnen >

integral of (4x^2-21x)/((x-3)^2(2x+3))

Lösung

\int \frac{4 x ^{2} - 21 x}{( x - 3 ) ^{2} ( 2 x + 3 )} d x

Lösung

ln ∣ x - 3 ∣ + \frac{3}{x - 3} + ln ∣ 2 x + 3 ∣ + C

Schritte zur Lösung

\int \frac{4 x ^{2} - 21 x}{( x - 3 ) ^{2} ( 2 x + 3 )} d x

Ermittle den Partialbruch von

\frac{4 x ^{2} - 21 x}{( x - 3 ) ^{2} ( 2 x + 3 )} : \frac{1}{x - 3} - \frac{3}{( x - 3 ) ^{2}} + \frac{2}{2 x + 3}

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

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

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

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

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

Vereinfache

= ln ∣ x - 3 ∣ + \frac{3}{x - 3} + ln ∣ 2 x + 3 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x - 3 ∣ + \frac{3}{x - 3} + ln ∣ 2 x + 3 ∣ + C

Graph

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