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

Lösung

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

- \frac{1}{x} + ln ∣ x ∣ - \frac{2}{x} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= - \frac{1}{x} + ln ∣ x ∣ - \frac{2}{x}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{x} + ln ∣ x ∣ - \frac{2}{x} + C