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

Lösung

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

\frac{x ^{5}}{5} + x^{2} - \frac{1}{x} + C

Schritte zur Lösung

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

Multipliziere aus

(x^{2} + \frac{1}{x})^{2} : x^{4} + 2 x + \frac{1}{x ^{2}}

= \int x^{4} + 2 x + \frac{1}{x ^{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 x^{4} d x + \int 2 x d x + \int \frac{1}{x ^{2}} d x

\int x^{4} d x = \frac{x ^{5}}{5}

\int 2 x d x = x^{2}

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

= \frac{x ^{5}}{5} + x^{2} - \frac{1}{x}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{5}}{5} + x^{2} - \frac{1}{x} + C

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

\int \frac{x ^{3}}{x ^{2} - 25} d x

x \to 0 lim (\frac{x ^{2} - 6}{x})

n or ma l f (x) = \frac{x}{1 + x ^{2}}, (2, \frac{2}{5})

\int \frac{2 x}{x ^{2} - 2 x + 5} d x