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Beliebt Rechnen >

integral of (8x^2+5x+8)/((x^2+1)^2)

Lösung

\int \frac{8 x ^{2} + 5 x + 8}{( x ^{2} + 1 ) ^{2}} d x

Lösung

8 arctan (x) - \frac{5}{2 ( x ^{2} + 1 )} + C

Schritte zur Lösung

\int \frac{8 x ^{2} + 5 x + 8}{( x ^{2} + 1 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{8 x ^{2} + 5 x + 8}{( x ^{2} + 1 ) ^{2}} : \frac{8}{x ^{2} + 1} + \frac{5 x}{( x ^{2} + 1 ) ^{2}}

= \int \frac{8}{x ^{2} + 1} + \frac{5 x}{( x ^{2} + 1 ) ^{2}} 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

= \int \frac{8}{x ^{2} + 1} d x + \int \frac{5 x}{( x ^{2} + 1 ) ^{2}} d x

\int \frac{8}{x ^{2} + 1} d x = 8 arctan (x)

\int \frac{5 x}{( x ^{2} + 1 ) ^{2}} d x = - \frac{5}{2 ( x ^{2} + 1 )}

= 8 arctan (x) - \frac{5}{2 ( x ^{2} + 1 )}

Füge eine Konstante zur Lösung hinzu

= 8 arctan (x) - \frac{5}{2 ( x ^{2} + 1 )} + C

Graph

Beliebte Beispiele

integral of 2sin(x)

\int 2 sin (x) d x

integral of (e^{5x})/(1+e^{5x)}

\int \frac{e ^{5 x}}{1 + e ^{5 x}} d x

integral of (x^2y^2)

\int (x^{2} y^{2}) d x

derivative x^2-5x+1/(x^{3/2)}

d er i v a t i v e x^{2} - 5 x + \frac{1}{x ^{\frac{3}{2}}}

limit as x approaches 3 of (x-3)/(1+x)

x \to 3 lim (\frac{x - 3}{1 + x})