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

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{6 x + 5}{x ^{2}} d x

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