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Beliebt Rechnen >

integral of 1/(ax^2+x)

Lösung

\int \frac{1}{a x ^{2} + x} d x

Lösung

- ln ∣ a x + 1 ∣ + ln (∣ a ∣ ∣ x ∣) + C

Schritte zur Lösung

\int \frac{1}{a x ^{2} + x} d x

Faktorisiere

\frac{1}{a x ^{2} + x}

= \int \frac{1}{x ( a x + 1 )} d x

Wende U-Substitution an

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

Ermittle den Partialbruch von

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

= \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

= - \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 ∣

= - ln ∣ u ∣ + ln ∣ u - 1 ∣

Setze in

u = a x + 1

ein

= - ln ∣ a x + 1 ∣ + ln ∣ a x + 1 - 1 ∣

ln ∣ a x + 1 - 1 ∣ = ln (∣ a ∣ ∣ x ∣)

= - ln ∣ a x + 1 ∣ + ln (∣ a ∣ ∣ x ∣)

Füge eine Konstante zur Lösung hinzu

= - ln ∣ a x + 1 ∣ + ln (∣ a ∣ ∣ x ∣) + C

Graph

Beliebte Beispiele

integral of 6x^2sqrt((2x^3))

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

integral of (6x^2-3x+5)/((2x-1)(x^2+1))

\int \frac{6 x ^{2} - 3 x + 5}{( 2 x - 1 ) ( x ^{2} + 1 )} d x

integral of (2x^2)/(x+6)

\int \frac{2 x ^{2}}{x + 6} d x

integral of sin^2(x)+3

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

integral of 1/(x^4sqrt(4x^2-9))

\int \frac{1}{x ^{4} 4 x ^{2} - 9} d x