integral of 1/(x^{1/2)-4}

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{2 u}{u - 4} d u

Entferne die Konstante:

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

= 2 \cdot \int \frac{u}{u - 4} d u

Wende U-Substitution an

= 2 \cdot \int \frac{v + 4}{v} d v

Multipliziere aus

\frac{v + 4}{v} : 1 + \frac{4}{v}

= 2 \cdot \int 1 + \frac{4}{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

= 2 (\int 1 d v + \int \frac{4}{v} d v)

\int 1 d v = v

\int \frac{4}{v} d v = 4 ln ∣ v ∣

= 2 (v + 4 ln ∣ v ∣)

Ersetze zurück

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

Füge eine Konstante zur Lösung hinzu

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

\int x^{2} 5 - 2 x^{3} d x

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