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Beliebt Rechnen >

integral of (5x^2-4x+4)/(x^3+4x)

Lösung

\int \frac{5 x ^{2} - 4 x + 4}{x ^{3} + 4 x} d x

Lösung

2 ln x^{2} + 4 - 2 arctan (\frac{x}{2}) + ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{5 x ^{2} - 4 x + 4}{x ^{3} + 4 x} d x

Multipliziere aus

\frac{5 x ^{2} - 4 x + 4}{x ^{3} + 4 x} : \frac{5 x ^{2}}{x ^{3} + 4 x} - \frac{4 x}{x ^{3} + 4 x} + \frac{4}{x ^{3} + 4 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} + 4 x} d x - \int \frac{4 x}{x ^{3} + 4 x} d x + \int \frac{4}{x ^{3} + 4 x} d x

\int \frac{5 x ^{2}}{x ^{3} + 4 x} d x = \frac{5}{2} ln x^{2} + 4

\int \frac{4 x}{x ^{3} + 4 x} d x = 2 arctan (\frac{x}{2})

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

= \frac{5}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2}) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 4

Vereinfache

\frac{5}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2}) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 4 : 2 ln x^{2} + 4 - 2 arctan (\frac{x}{2}) + ln ∣ x ∣

= 2 ln x^{2} + 4 - 2 arctan (\frac{x}{2}) + ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= 2 ln x^{2} + 4 - 2 arctan (\frac{x}{2}) + ln ∣ x ∣ + C

Graph

Beliebte Beispiele

integral from 2 to 7 of t^2ln(3t)

\int_{2}^{7} t^{2} ln (3 t) d t

integral of 1/((x-x^2))

\int \frac{1}{( x - x ^{2} )} d x

y^'+x/(x^2+1)y=-4x

y^{^{'}} + \frac{x}{x ^{2} + 1} y = - 4 x

tangent y=(((x+2))/((x-3)))^2,\at x=4

t an g e n t y = (\frac{( x + 2 )}{( x - 3 )})^{2}, at x = 4

derivative of a(e^x+e^xx)

\frac{d}{d x} (a (e^{x} + e^{x} x))