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Beliebt Rechnen >

integral of 1/(x+3sqrt(x)-4)

Lösung

\int \frac{1}{x + 3 x - 4} d x

Lösung

2 (\frac{1}{5} ln x - 1 + \frac{4}{5} ln x + 4) + C

Schritte zur Lösung

\int \frac{1}{x + 3 x - 4} d x

Wende U-Substitution an

= \int \frac{2 u}{u ^{2} + 3 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 ^{2} + 3 u - 4} d u

Ermittle den Partialbruch von

\frac{u}{u ^{2} + 3 u - 4} : \frac{1}{5 ( u - 1 )} + \frac{4}{5 ( u + 4 )}

= 2 \cdot \int \frac{1}{5 ( u - 1 )} + \frac{4}{5 ( 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

= 2 (\int \frac{1}{5 ( u - 1 )} d u + \int \frac{4}{5 ( u + 4 )} d u)

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

\int \frac{4}{5 ( u + 4 )} d u = \frac{4}{5} ln ∣ u + 4 ∣

= 2 (\frac{1}{5} ln ∣ u - 1 ∣ + \frac{4}{5} ln ∣ u + 4 ∣)

Setze in

u = x

ein

= 2 (\frac{1}{5} ln x - 1 + \frac{4}{5} ln x + 4)

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{5} ln x - 1 + \frac{4}{5} ln x + 4) + C

Graph

Beliebte Beispiele

integral of 1/2 cot^2(x)csc^4(x)

\int \frac{1}{2} cot^{2} (x) csc^{4} (x) d x

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