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

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{( e ^{2 x} )}{4 + e ^{4 x}} d x

\int l d l

x \to 1 lim (\frac{9}{x ^{3} - 1})

\int_{0}^{3} 2 π x (3 x - x^{2}) d x

\frac{\partial}{\partial x} (cos (5 x + 9 y + 4 z))