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Beliebt Rechnen >

integral of (40x^2)/(x^4-52x^2+576)

Lösung

\int \frac{40 x ^{2}}{x ^{4} - 52 x ^{2} + 576} d x

Lösung

40 (\frac{1}{10} ln ∣ x + 4 ∣ - \frac{1}{10} ln ∣ x - 4 ∣ - \frac{3}{20} ln ∣ x + 6 ∣ + \frac{3}{20} ln ∣ x - 6 ∣) + C

Schritte zur Lösung

\int \frac{40 x ^{2}}{x ^{4} - 52 x ^{2} + 576} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 40 \cdot \int \frac{x ^{2}}{x ^{4} - 52 x ^{2} + 576} d x

Ermittle den Partialbruch von

\frac{x ^{2}}{x ^{4} - 52 x ^{2} + 576} : \frac{1}{10 ( x + 4 )} - \frac{1}{10 ( x - 4 )} - \frac{3}{20 ( x + 6 )} + \frac{3}{20 ( x - 6 )}

= 40 \cdot \int \frac{1}{10 ( x + 4 )} - \frac{1}{10 ( x - 4 )} - \frac{3}{20 ( x + 6 )} + \frac{3}{20 ( x - 6 )} 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

= 40 (\int \frac{1}{10 ( x + 4 )} d x - \int \frac{1}{10 ( x - 4 )} d x - \int \frac{3}{20 ( x + 6 )} d x + \int \frac{3}{20 ( x - 6 )} d x)

\int \frac{1}{10 ( x + 4 )} d x = \frac{1}{10} ln ∣ x + 4 ∣

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

\int \frac{3}{20 ( x + 6 )} d x = \frac{3}{20} ln ∣ x + 6 ∣

\int \frac{3}{20 ( x - 6 )} d x = \frac{3}{20} ln ∣ x - 6 ∣

= 40 (\frac{1}{10} ln ∣ x + 4 ∣ - \frac{1}{10} ln ∣ x - 4 ∣ - \frac{3}{20} ln ∣ x + 6 ∣ + \frac{3}{20} ln ∣ x - 6 ∣)

Füge eine Konstante zur Lösung hinzu

= 40 (\frac{1}{10} ln ∣ x + 4 ∣ - \frac{1}{10} ln ∣ x - 4 ∣ - \frac{3}{20} ln ∣ x + 6 ∣ + \frac{3}{20} ln ∣ x - 6 ∣) + C

Graph

Beliebte Beispiele

steigung (2,3),(-2,-3)

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derivative of log_{10}(x/(sqrt(x^2+1)))

\frac{d}{d x} (lo g_{10} (\frac{x}{x ^{2} + 1}))

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\frac{\partial}{\partial x} (15 - 2 x^{2} - y^{2})