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Beliebt Rechnen >

integral of (28)/(4x+xsqrt(x))

Lösung

\int \frac{28}{4 x + x x} d x

Lösung

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

Schritte zur Lösung

\int \frac{28}{4 x + x x} d x

Entferne die Konstante:

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

= 28 \cdot \int \frac{1}{4 x + x x} d x

Wende U-Substitution an

= 28 \cdot \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

= 28 \cdot 2 \cdot \int \frac{1}{u ( u + 4 )} d u

Ermittle den Partialbruch von

\frac{1}{u ( u + 4 )} : \frac{1}{4 u} - \frac{1}{4 ( u + 4 )}

= 28 \cdot 2 \cdot \int \frac{1}{4 u} - \frac{1}{4 ( u + 4 )} 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

= 28 \cdot 2 (\int \frac{1}{4 u} d u - \int \frac{1}{4 ( u + 4 )} d u)

\int \frac{1}{4 u} d u = \frac{1}{4} ln ∣ u ∣

\int \frac{1}{4 ( u + 4 )} d u = \frac{1}{4} ln ∣ u + 4 ∣

= 28 \cdot 2 (\frac{1}{4} ln ∣ u ∣ - \frac{1}{4} ln ∣ u + 4 ∣)

Setze in

u = x

ein

= 28 \cdot 2 (\frac{1}{4} ln x - \frac{1}{4} ln x + 4)

Vereinfache

= 56 (\frac{1}{4} ln x - \frac{1}{4} ln x + 4)

Füge eine Konstante zur Lösung hinzu

= 56 (\frac{1}{4} ln x - \frac{1}{4} ln x + 4) + C

Graph

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