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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{x ^{3}}{3} + 5 x - 6 ln \frac{x}{2} + 1 + 6 ln \frac{x}{2} - 1 + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{4} + x ^{2} + 4}{x ^{2} - 4} : \frac{x ^{4}}{x ^{2} - 4} + \frac{x ^{2}}{x ^{2} - 4} + \frac{4}{x ^{2} - 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{x ^{4}}{x ^{2} - 4} d x + \int \frac{x ^{2}}{x ^{2} - 4} d x + \int \frac{4}{x ^{2} - 4} d x

\int \frac{x ^{4}}{x ^{2} - 4} d x = - 8 (- \frac{x ^{3}}{24} - \frac{x}{2} + \frac{1}{2} ln \frac{x}{2} + 1 - \frac{1}{2} ln \frac{x}{2} - 1)

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

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

= - 8 (- \frac{x ^{3}}{24} - \frac{x}{2} + \frac{1}{2} ln \frac{x}{2} + 1 - \frac{1}{2} ln \frac{x}{2} - 1) - ln \frac{x}{2} + 1 + ln \frac{x}{2} - 1 + x - ln \frac{x}{2} + 1 + ln \frac{x}{2} - 1

Vereinfache

- 8 (- \frac{x ^{3}}{24} - \frac{x}{2} + \frac{1}{2} ln \frac{x}{2} + 1 - \frac{1}{2} ln \frac{x}{2} - 1) - ln \frac{x}{2} + 1 + ln \frac{x}{2} - 1 + x - ln \frac{x}{2} + 1 + ln \frac{x}{2} - 1 : \frac{x ^{3}}{3} + 5 x - 6 ln \frac{x}{2} + 1 + 6 ln \frac{x}{2} - 1

= \frac{x ^{3}}{3} + 5 x - 6 ln \frac{x}{2} + 1 + 6 ln \frac{x}{2} - 1

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} + 5 x - 6 ln \frac{x}{2} + 1 + 6 ln \frac{x}{2} - 1 + C

Graph

Beliebte Beispiele

integral of (4x^2)/(sqrt(x^3+16))

\int \frac{4 x ^{2}}{x ^{3} + 16} d x

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y^{^{'}} = t^{2} y^{3}