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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Vervollständige das Quadrat

x^{2} + 4 x + 3 : (x + 2)^{2} - 1

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

Wende U-Substitution an

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

Multipliziere aus

\frac{u - 2}{u ^{2} - 1} : \frac{u}{u ^{2} - 1} - \frac{2}{u ^{2} - 1}

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 ^{2} - 1} d u - \int \frac{2}{u ^{2} - 1} d u

\int \frac{u}{u ^{2} - 1} d u = \frac{1}{2} ln u^{2} - 1

\int \frac{2}{u ^{2} - 1} d u = - ln ∣ u + 1 ∣ + ln ∣ u - 1 ∣

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

Setze in

u = x + 2

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 1/(4-y)

\int \frac{1}{4 - y} d y

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x \to 8 + lim (\frac{x - 8}{∣ x - 8 ∣})

tangent y=3x^2-2x,\at x=-1

t an g e n t y = 3 x^{2} - 2 x, at x = - 1

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\frac{d}{d x} (x^{2} (x - 2)^{4})

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\frac{\partial}{\partial x} (2 x + 3 y)