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

Lösung

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

- \frac{1}{x} + 4 x + \frac{x ^{2}}{2} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

= - \frac{1}{x} + 4 x^{\frac{1}{2}} + \frac{x ^{2}}{2}

Wende Radikal Regel an:

a^{\frac{1}{n}} = n a

= - \frac{1}{x} + 4 x + \frac{x ^{2}}{2}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{x} + 4 x + \frac{x ^{2}}{2} + C

\int \frac{6 x - 3}{4 x ^{2} - 4 x + 1} d x

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

\int \frac{x ^{5} - 5 x + 4}{x ^{2}} d x

\int \frac{4 x}{5 x ^{2} + 2} d x

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