integral of x/(x^{2/3)+x^{8/6}}

Lösung

\int \frac{x}{x ^{\frac{2}{3}} + x ^{\frac{8}{6}}} d x

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

Schritte zur Lösung

\int \frac{x}{x ^{\frac{2}{3}} + x ^{\frac{8}{6}}} d x

Wende U-Substitution an

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{3}{2} \cdot \int \frac{u}{u + 1} d u

Wende U-Substitution an

= \frac{3}{2} \cdot \int \frac{v - 1}{v} d v

Multipliziere aus

\frac{v - 1}{v} : 1 - \frac{1}{v}

= \frac{3}{2} \cdot \int 1 - \frac{1}{v} d v

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{3}{2} (\int 1 d v - \int \frac{1}{v} d v)

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= \frac{3}{2} (v - ln ∣ v ∣)

Ersetze zurück

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{sec ( θ ) tan ( θ )}{3 + 2 sec ( θ )} d θ

\int (3 - 0.01 x + 0.00006 x^{2}) d x

\int \frac{1}{( x + 2 ) x ^{2} + 4 x - 5} d x

\int 2 cos (x) (\frac{2 csc ( x )}{cot ( x )}) d x

\int \frac{1}{x ( 4 + ( 3 ln ( x ) ) ^{2} )} d x