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Beliebt Rechnen >

integral of (x^2+5x+7)/(x+3)

Lösung

\int \frac{x ^{2} + 5 x + 7}{x + 3} d x

Lösung

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

Schritte zur Lösung

\int \frac{x ^{2} + 5 x + 7}{x + 3} d x

Wende U-Substitution an

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

Multipliziere aus

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

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

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

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{u ^{2}}{2} - u + ln ∣ u ∣

Setze in

u = x + 3

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

inverselaplace (4se^{-3s})/((s^2+4))

in v erse l a pl a ce \frac{4 s e ^{- 3 s}}{( s ^{2} + 4 )}

implicit (dy)/(dx),y=2x-5

im pl i c i t \frac{d y}{d x}, y = 2 x - 5

sum from n=1 to infinity of arctan(4n)

n = 1 \sum \infty arctan (4 n)

inverselaplace (2s^2+1)/(s+3)

in v erse l a pl a ce \frac{2 s ^{2} + 1}{s + 3}

(xdy)/(dx)=y+xe^{-y/x}

\frac{x d y}{d x} = y + x e^{- \frac{y}{x}}