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Beliebt Rechnen >

integral of (x-1)/((x+1)(x+4))

Lösung

\int \frac{x - 1}{( x + 1 ) ( x + 4 )} d x

Lösung

ln ∣ x + 4 ∣ - 2 (\frac{1}{3} ln ∣ x + 1 ∣ - \frac{1}{3} ln ∣ x + 4 ∣) + C

Schritte zur Lösung

\int \frac{x - 1}{( x + 1 ) ( x + 4 )} d x

Wende U-Substitution an

= \int \frac{u - 2}{u ( u + 3 )} d u

Multipliziere aus

\frac{u - 2}{u ( u + 3 )} : \frac{u}{u ( u + 3 )} - \frac{2}{u ( u + 3 )}

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{u}{u ( u + 3 )} d u - \int \frac{2}{u ( u + 3 )} d u

\int \frac{u}{u ( u + 3 )} d u = ln ∣ u + 3 ∣

\int \frac{2}{u ( u + 3 )} d u = 2 (\frac{1}{3} ln ∣ u ∣ - \frac{1}{3} ln ∣ u + 3 ∣)

= ln ∣ u + 3 ∣ - 2 (\frac{1}{3} ln ∣ u ∣ - \frac{1}{3} ln ∣ u + 3 ∣)

Setze in

u = x + 1

ein

= ln ∣ x + 1 + 3 ∣ - 2 (\frac{1}{3} ln ∣ x + 1 ∣ - \frac{1}{3} ln ∣ x + 1 + 3 ∣)

Vereinfache

ln ∣ x + 1 + 3 ∣ - 2 (\frac{1}{3} ln ∣ x + 1 ∣ - \frac{1}{3} ln ∣ x + 1 + 3 ∣) : ln ∣ x + 4 ∣ - 2 (\frac{1}{3} ln ∣ x + 1 ∣ - \frac{1}{3} ln ∣ x + 4 ∣)

= ln ∣ x + 4 ∣ - 2 (\frac{1}{3} ln ∣ x + 1 ∣ - \frac{1}{3} ln ∣ x + 4 ∣)

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 4 ∣ - 2 (\frac{1}{3} ln ∣ x + 1 ∣ - \frac{1}{3} ln ∣ x + 4 ∣) + C

Graph

Beliebte Beispiele

integral of 1/(64-v^2)

\int \frac{1}{64 - v ^{2}} d v

integral of 7x^3e^{x^2}

\int 7 x^{3} e^{x^{2}} d x

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

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

integral of ((ln(x))^2)/((x))

\int \frac{( ln ( x ) ) ^{2}}{( x )} d x

integral of (3^{2x})/(4+3^{2x)}

\int \frac{3 ^{2 x}}{4 + 3 ^{2 x}} d x