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Beliebt Rechnen >

integral of x/(5x-2)

Lösung

\int \frac{x}{5 x - 2} d x

Lösung

\frac{1}{25} (5 x - 2 + 2 ln ∣ 5 x - 2 ∣) + C

Schritte zur Lösung

\int \frac{x}{5 x - 2} d x

Wende U-Substitution an

= \int \frac{u + 2}{25 u} d u

Entferne die Konstante:

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

= \frac{1}{25} \cdot \int \frac{u + 2}{u} d u

Multipliziere aus

\frac{u + 2}{u} : 1 + \frac{2}{u}

= \frac{1}{25} \cdot \int 1 + \frac{2}{u} 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

= \frac{1}{25} (\int 1 d u + \int \frac{2}{u} d u)

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= \frac{1}{25} (u + 2 ln ∣ u ∣)

Setze in

u = 5 x - 2

ein

= \frac{1}{25} (5 x - 2 + 2 ln ∣ 5 x - 2 ∣)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{25} (5 x - 2 + 2 ln ∣ 5 x - 2 ∣) + C

Graph

Beliebte Beispiele

integral from 1/3 to 7 of 6xln(3x)

\int_{\frac{1}{3}}^{7} 6 x ln (3 x) d x

derivative of (4x^3+2x+5/x)

\frac{d}{d x} (\frac{4 x ^{3} + 2 x + 5}{x})

y^{^{'}} sin (y) = x^{4}

limit as x approaches 3 of sqrt(6x-2)

x \to 3 lim (6 x - 2)

integral of (x+9)/(sqrt(x))

\int \frac{x + 9}{x} d x