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Beliebt Rechnen >

integral of (5x^2-2x+64)/(x^3-16x)

Lösung

\int \frac{5 x ^{2} - 2 x + 64}{x ^{3} - 16 x} d x

Lösung

\frac{5}{2} ln (∣ x + 4 ∣ ∣ x - 4 ∣) + \frac{9}{4} ln ∣ x + 4 ∣ + \frac{7}{4} ln ∣ x - 4 ∣ - 4 ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{5 x ^{2} - 2 x + 64}{x ^{3} - 16 x} d x

Multipliziere aus

\frac{5 x ^{2} - 2 x + 64}{x ^{3} - 16 x} : \frac{5 x ^{2}}{x ^{3} - 16 x} - \frac{2 x}{x ^{3} - 16 x} + \frac{64}{x ^{3} - 16 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{5 x ^{2}}{x ^{3} - 16 x} d x - \int \frac{2 x}{x ^{3} - 16 x} d x + \int \frac{64}{x ^{3} - 16 x} d x

\int \frac{5 x ^{2}}{x ^{3} - 16 x} d x = \frac{5}{2} ln (∣ x + 4 ∣ ∣ x - 4 ∣)

\int \frac{2 x}{x ^{3} - 16 x} d x = 2 (- \frac{1}{8} ln ∣ x + 4 ∣ + \frac{1}{8} ln ∣ x - 4 ∣)

\int \frac{64}{x ^{3} - 16 x} d x = - 64 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{32} ln ∣ x + 4 ∣ - \frac{1}{32} ln ∣ x - 4 ∣)

= \frac{5}{2} ln (∣ x + 4 ∣ ∣ x - 4 ∣) - 2 (- \frac{1}{8} ln ∣ x + 4 ∣ + \frac{1}{8} ln ∣ x - 4 ∣) - 64 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{32} ln ∣ x + 4 ∣ - \frac{1}{32} ln ∣ x - 4 ∣)

Vereinfache

\frac{5}{2} ln (∣ x + 4 ∣ ∣ x - 4 ∣) - 2 (- \frac{1}{8} ln ∣ x + 4 ∣ + \frac{1}{8} ln ∣ x - 4 ∣) - 64 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{32} ln ∣ x + 4 ∣ - \frac{1}{32} ln ∣ x - 4 ∣) : \frac{5}{2} ln (∣ x + 4 ∣ ∣ x - 4 ∣) + \frac{9}{4} ln ∣ x + 4 ∣ + \frac{7}{4} ln ∣ x - 4 ∣ - 4 ln ∣ x ∣

= \frac{5}{2} ln (∣ x + 4 ∣ ∣ x - 4 ∣) + \frac{9}{4} ln ∣ x + 4 ∣ + \frac{7}{4} ln ∣ x - 4 ∣ - 4 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= \frac{5}{2} ln (∣ x + 4 ∣ ∣ x - 4 ∣) + \frac{9}{4} ln ∣ x + 4 ∣ + \frac{7}{4} ln ∣ x - 4 ∣ - 4 ln ∣ x ∣ + C

Graph

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