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

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

\int_{a}^{x} \frac{1}{b - a} d x

\frac{d}{d x} (\frac{cos ( 2 x ) + x ^{3}}{tan ( 2 x ) - 3 x})

x \to 2 - lim (\frac{x + 5}{x - 2})

in v erse l a pl a ce - \frac{e ^{- π s}}{s ^{2} + 1}

s l o p e (1950, 38300), (2000, 82500)