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Beliebt Rechnen >

integral of (272x^2+4)/(x^2(x-4))

Lösung

\int \frac{272 x ^{2} + 4}{x ^{2} ( x - 4 )} d x

Lösung

- \frac{1}{4} ln ∣ x ∣ + \frac{1089}{4} ln ∣ x - 4 ∣ + \frac{1}{x} + C

Schritte zur Lösung

\int \frac{272 x ^{2} + 4}{x ^{2} ( x - 4 )} d x

Multipliziere aus

\frac{272 x ^{2} + 4}{x ^{2} ( x - 4 )} : \frac{272 x ^{2}}{x ^{2} ( x - 4 )} + \frac{4}{x ^{2} ( x - 4 )}

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

\int \frac{272 x ^{2}}{x ^{2} ( x - 4 )} d x = 272 ln ∣ x - 4 ∣

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

= 272 ln ∣ x - 4 ∣ - \frac{1}{4} ln ∣ x ∣ + \frac{1}{x} + \frac{1}{4} ln ∣ x - 4 ∣

Vereinfache

272 ln ∣ x - 4 ∣ - \frac{1}{4} ln ∣ x ∣ + \frac{1}{x} + \frac{1}{4} ln ∣ x - 4 ∣ : - \frac{1}{4} ln ∣ x ∣ + \frac{1089}{4} ln ∣ x - 4 ∣ + \frac{1}{x}

= - \frac{1}{4} ln ∣ x ∣ + \frac{1089}{4} ln ∣ x - 4 ∣ + \frac{1}{x}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{4} ln ∣ x ∣ + \frac{1089}{4} ln ∣ x - 4 ∣ + \frac{1}{x} + C

Graph

Beliebte Beispiele

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integral of (9x)/(3x+1)

\int \frac{9 x}{3 x + 1} d x