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Beliebt Rechnen >

integral of (4x-11)/((x-5)(2x-1))

Lösung

\int \frac{4 x - 11}{( x - 5 ) ( 2 x - 1 )} d x

Lösung

ln ∣ 2 x - 1 ∣ + ln ∣ 2 x - 10 ∣ + C

Schritte zur Lösung

\int \frac{4 x - 11}{( x - 5 ) ( 2 x - 1 )} d x

Multipliziere aus

\frac{4 x - 11}{( x - 5 ) ( 2 x - 1 )} : \frac{4 x}{( x - 5 ) ( 2 x - 1 )} - \frac{11}{( x - 5 ) ( 2 x - 1 )}

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{4 x}{( x - 5 ) ( 2 x - 1 )} d x - \int \frac{11}{( x - 5 ) ( 2 x - 1 )} d x

\int \frac{4 x}{( x - 5 ) ( 2 x - 1 )} d x = 2 (\frac{10}{9} ln ∣ 2 x - 10 ∣ - \frac{1}{9} ln ∣ 2 x - 1 ∣)

\int \frac{11}{( x - 5 ) ( 2 x - 1 )} d x = 11 (- \frac{1}{9} ln ∣ 2 x - 1 ∣ + \frac{1}{9} ln ∣ 2 x - 10 ∣)

= 2 (\frac{10}{9} ln ∣ 2 x - 10 ∣ - \frac{1}{9} ln ∣ 2 x - 1 ∣) - 11 (- \frac{1}{9} ln ∣ 2 x - 1 ∣ + \frac{1}{9} ln ∣ 2 x - 10 ∣)

Vereinfache

2 (\frac{10}{9} ln ∣ 2 x - 10 ∣ - \frac{1}{9} ln ∣ 2 x - 1 ∣) - 11 (- \frac{1}{9} ln ∣ 2 x - 1 ∣ + \frac{1}{9} ln ∣ 2 x - 10 ∣) : ln ∣ 2 x - 1 ∣ + ln ∣ 2 x - 10 ∣

= ln ∣ 2 x - 1 ∣ + ln ∣ 2 x - 10 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ 2 x - 1 ∣ + ln ∣ 2 x - 10 ∣ + C

Graph

Beliebte Beispiele

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

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\int cos (6 x) cos (7 x) d x

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\int \frac{10}{5 x ^{2} - 2 x ^{3}} d x