integral of (2x^3+x+3)/((x^2+1)^2)

Lösung

\int \frac{2 x ^{3} + x + 3}{( x ^{2} + 1 ) ^{2}} d x

ln x^{2} + 1 + \frac{1}{x ^{2} + 1} - \frac{1}{2 ( x ^{2} + 1 )} + \frac{3}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) + C

Schritte zur Lösung

\int \frac{2 x ^{3} + x + 3}{( x ^{2} + 1 ) ^{2}} d x

Multipliziere aus

\frac{2 x ^{3} + x + 3}{( x ^{2} + 1 ) ^{2}} : \frac{2 x ^{3}}{( x ^{2} + 1 ) ^{2}} + \frac{x}{( x ^{2} + 1 ) ^{2}} + \frac{3}{( x ^{2} + 1 ) ^{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} + 1 ) ^{2}} d x + \int \frac{x}{( x ^{2} + 1 ) ^{2}} d x + \int \frac{3}{( x ^{2} + 1 ) ^{2}} d x

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

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

\int \frac{3}{( x ^{2} + 1 ) ^{2}} d x = \frac{3}{2} (arctan (x) + \frac{x}{1 + x ^{2}})

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

Füge eine Konstante zur Lösung hinzu

= ln x^{2} + 1 + \frac{1}{x ^{2} + 1} - \frac{1}{2 ( x ^{2} + 1 )} + \frac{3}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) + C

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