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Beliebt Rechnen >

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

Lösung

\int \frac{2 x ^{3} + 5 x}{( x ^{2} + 3 ) ^{2}} d x

Lösung

ln x^{2} + 3 + \frac{3}{x ^{2} + 3} - \frac{5}{2 ( x ^{2} + 3 )} + C

Schritte zur Lösung

\int \frac{2 x ^{3} + 5 x}{( x ^{2} + 3 ) ^{2}} d x

Multipliziere aus

\frac{2 x ^{3} + 5 x}{( x ^{2} + 3 ) ^{2}} : \frac{2 x ^{3}}{( x ^{2} + 3 ) ^{2}} + \frac{5 x}{( x ^{2} + 3 ) ^{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 ^{3}}{( x ^{2} + 3 ) ^{2}} d x + \int \frac{5 x}{( x ^{2} + 3 ) ^{2}} d x

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

\int \frac{5 x}{( x ^{2} + 3 ) ^{2}} d x = - \frac{5}{2 ( x ^{2} + 3 )}

= ln x^{2} + 3 + \frac{3}{x ^{2} + 3} - \frac{5}{2 ( x ^{2} + 3 )}

Füge eine Konstante zur Lösung hinzu

= ln x^{2} + 3 + \frac{3}{x ^{2} + 3} - \frac{5}{2 ( x ^{2} + 3 )} + C

Graph

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