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Beliebt Rechnen >

integral of x/(a+bx)

Lösung

\int \frac{x}{a + b x} d x

Lösung

\frac{1}{b ^{2}} (a + b x - a ln ∣ a + b x ∣) + C

Schritte zur Lösung

\int \frac{x}{a + b x} d x

Wende U-Substitution an

= \int \frac{u - a}{b ^{2} u} d u

Entferne die Konstante:

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

= \frac{1}{b ^{2}} \cdot \int \frac{u - a}{u} d u

Multipliziere aus

\frac{u - a}{u} : 1 - \frac{a}{u}

= \frac{1}{b ^{2}} \cdot \int 1 - \frac{a}{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}{b ^{2}} (\int 1 d u - \int \frac{a}{u} d u)

\int 1 d u = u

\int \frac{a}{u} d u = a ln ∣ u ∣

= \frac{1}{b ^{2}} (u - a ln ∣ u ∣)

Setze in

u = a + b x

ein

= \frac{1}{b ^{2}} (a + b x - a ln ∣ a + b x ∣)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{b ^{2}} (a + b x - a ln ∣ a + b x ∣) + C

Graph

Beliebte Beispiele

derivative of (x^2-6(x^3+x^2+x+2))

\frac{d}{d x} ((x^{2} - 6) (x^{3} + x^{2} + x + 2))

integral of 2(x-6)^3

\int 2 (x - 6)^{3} d x

geradzahligkeit 2x^2cos(x)cot(x)

p a r i t y 2 x^{2} cos (x) cot (x)

xy^'+y=-3xy^2,y(1)=-4

x y^{^{'}} + y = - 3 x y^{2}, y (1) = - 4

tangent y=4-x^2

t an g e n t y = 4 - x^{2}