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Beliebt Rechnen >

integral of x/(ax+b)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Multipliziere aus

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

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

\int 1 d u = u

\int \frac{b}{u} d u = b ln ∣ u ∣

= \frac{1}{a ^{2}} (u - b ln ∣ u ∣)

Setze in

u = a x + b

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral from-3 to-5 of (-5)^{-3}

\int_{- 3}^{- 5} (- 5)^{- 3} d x

derivative of-(y^2/((x-y)^2))

\frac{d}{d x} (- \frac{y ^{2}}{( x - y ) ^{2}})

derivative 2^{x^2}

d er i v a t i v e 2^{x^{2}}

integral of 1/((x^2+2)^{3/2)}

\int \frac{1}{( x ^{2} + 2 ) ^{\frac{3}{2}}} d x

integral of 1/(x^2+8x+25)

\int \frac{1}{x ^{2} + 8 x + 25} d x