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Beliebt Rechnen >

integral of 1/(x+xsqrt(x))

Lösung

\int \frac{1}{x + x x} d x

Lösung

2 (ln x - ln x + 1) + C

Schritte zur Lösung

\int \frac{1}{x + x x} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

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

= 2 (ln ∣ u ∣ - ln ∣ u + 1 ∣)

Setze in

u = x

ein

= 2 (ln x - ln x + 1)

Füge eine Konstante zur Lösung hinzu

= 2 (ln x - ln x + 1) + C

Graph

Beliebte Beispiele

integral of 8x+1/(sqrt(x))

\int 8 x + \frac{1}{x} d x

derivative of (x+4/(3x^2e^x))

\frac{d}{d x} (\frac{x + 4}{3 x ^{2} e ^{x}})

integral from 0 to 1 of e^{sqrt(x)}

\int_{0}^{1} e^{x} d x

derivative of sqrt(x+a)

\frac{d}{d x} (x + a)

derivative of ln(x-y)

\frac{d}{d x} (ln (x - y))