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Beliebt Rechnen >

integral of ((x^2-2x+1))/((x^2+x))

Lösung

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

Lösung

ln ∣ x ∣ - 4 ln ∣ x + 1 ∣ + x - 1 + C

Schritte zur Lösung

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

Vervollständige das Quadrat

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

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

Wende U-Substitution an

= \int \frac{u ^{2}}{u ^{2} + 3 u + 2} d u

\frac{u ^{2}}{u ^{2} + 3 u + 2} = \frac{- 3 u - 2}{u ^{2} + 3 u + 2} + 1

= \int \frac{- 3 u - 2}{u ^{2} + 3 u + 2} + 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{- 3 u - 2}{u ^{2} + 3 u + 2} d u + \int 1 d u

\int \frac{- 3 u - 2}{u ^{2} + 3 u + 2} d u = ln ∣ u + 1 ∣ - 4 ln ∣ u + 2 ∣

\int 1 d u = u

= ln ∣ u + 1 ∣ - 4 ln ∣ u + 2 ∣ + u

Setze in

u = x - 1

ein

= ln ∣ x - 1 + 1 ∣ - 4 ln ∣ x - 1 + 2 ∣ + x - 1

Vereinfache

= ln ∣ x ∣ - 4 ln ∣ x + 1 ∣ + x - 1

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ - 4 ln ∣ x + 1 ∣ + x - 1 + C

Graph

Beliebte Beispiele

derivative of 1/(a-x)

\frac{d}{d x} (\frac{1}{a - x})

(\partial)/(\partial x)(-0.06x^2+1.08x)

\frac{\partial}{\partial x} (- 0.06 x^{2} + 1.08 x)

derivative 2x^3-x^2+2

d er i v a t i v e 2 x^{3} - x^{2} + 2

integral of (6x^2-3x+4)

\int (6 x^{2} - 3 x + 4) d x

limit as x approaches 1/2 of xsec(pi)x

x \to \frac{1}{2} lim (x sec (π) x)