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

Lösung

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

ln ∣ x ∣ + \frac{2}{x + 1} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ - (- \frac{2}{x + 1})

Vereinfache

= ln ∣ x ∣ + \frac{2}{x + 1}

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ + \frac{2}{x + 1} + C

\int (\frac{2 x - 1}{( x + 1 ) ( x ^{2} + 4 x + 4 )}) d x

\int \frac{49}{x ^{2} 49 - x ^{2}} d x

\int r sin (r) d r

\int \frac{x + 8}{x ^{3} + 13 x ^{2} + 35 x - 49} d x

\int 0.06 e^{0.02 x} d x