integral of (15)/(5x+xsqrt(x))

Lösung

\int \frac{15}{5 x + x x} d x

30 (\frac{1}{5} ln x - \frac{1}{5} ln x + 5) + C

Schritte zur Lösung

\int \frac{15}{5 x + x x} d x

Entferne die Konstante:

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

= 15 \cdot \int \frac{1}{5 x + x x} d x

Wende U-Substitution an

= 15 \cdot \int \frac{2}{u ( u + 5 )} d u

Entferne die Konstante:

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

= 15 \cdot 2 \cdot \int \frac{1}{u ( u + 5 )} d u

Ermittle den Partialbruch von

\frac{1}{u ( u + 5 )} : \frac{1}{5 u} - \frac{1}{5 ( u + 5 )}

= 15 \cdot 2 \cdot \int \frac{1}{5 u} - \frac{1}{5 ( u + 5 )} d u

Wende die Summenregel an:

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

= 15 \cdot 2 (\int \frac{1}{5 u} d u - \int \frac{1}{5 ( u + 5 )} d u)

\int \frac{1}{5 u} d u = \frac{1}{5} ln ∣ u ∣

\int \frac{1}{5 ( u + 5 )} d u = \frac{1}{5} ln ∣ u + 5 ∣

= 15 \cdot 2 (\frac{1}{5} ln ∣ u ∣ - \frac{1}{5} ln ∣ u + 5 ∣)

Setze in

u = x

ein

= 15 \cdot 2 (\frac{1}{5} ln x - \frac{1}{5} ln x + 5)

Vereinfache

= 30 (\frac{1}{5} ln x - \frac{1}{5} ln x + 5)

Füge eine Konstante zur Lösung hinzu

= 30 (\frac{1}{5} ln x - \frac{1}{5} ln x + 5) + C

x \to 0 + lim (\frac{x + 3}{x ^{2}})

d er i v a t i v e ln (\frac{5 x ^{2} + x + 6}{x ^{3}})

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