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Beliebt Rechnen >

integral of (2x^2+6x-16)/(2x+4)

Lösung

\int \frac{2 x ^{2} + 6 x - 16}{2 x + 4} d x

Lösung

\frac{1}{2} (x + 2)^{2} - x - 2 - 10 ln ∣ x + 2 ∣ + C

Schritte zur Lösung

\int \frac{2 x ^{2} + 6 x - 16}{2 x + 4} d x

Vereinfache

\frac{2 x ^{2} + 6 x - 16}{2 x + 4} : \frac{x ^{2} + 3 x - 8}{x + 2}

= \int \frac{x ^{2} + 3 x - 8}{x + 2} d x

Wende U-Substitution an

= \int \frac{u ^{2} - u - 10}{u} d u

Multipliziere aus

\frac{u ^{2} - u - 10}{u} : u - 1 - \frac{10}{u}

= \int u - 1 - \frac{10}{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

= \int u d u - \int 1 d u - \int \frac{10}{u} d u

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

\int 1 d u = u

\int \frac{10}{u} d u = 10 ln ∣ u ∣

= \frac{u ^{2}}{2} - u - 10 ln ∣ u ∣

Setze in

u = x + 2

ein

= \frac{( x + 2 ) ^{2}}{2} - (x + 2) - 10 ln ∣ x + 2 ∣

Vereinfache

\frac{( x + 2 ) ^{2}}{2} - (x + 2) - 10 ln ∣ x + 2 ∣ : \frac{1}{2} (x + 2)^{2} - x - 2 - 10 ln ∣ x + 2 ∣

= \frac{1}{2} (x + 2)^{2} - x - 2 - 10 ln ∣ x + 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (x + 2)^{2} - x - 2 - 10 ln ∣ x + 2 ∣ + C

Graph

Beliebte Beispiele

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

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

integral of 1/((3-x))

\int \frac{1}{( 3 - x )} d x

integral of (e^x)/(13+e^x)

\int \frac{e ^{x}}{13 + e ^{x}} d x

integral of x^{(-1)/5}

\int x^{\frac{- 1}{5}} d x

integral of 3/(x^2-5x+4)

\int \frac{3}{x ^{2} - 5 x + 4} d x