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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{x ^{2}}{8} + \frac{3}{4} x + \frac{1}{2} (- \frac{1}{8} ln 4 x^{2} + 2 + \frac{5}{2 2} arctan (2 x)) + C

Schritte zur Lösung

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

Lange Division

\frac{x ^{3} + 3 x ^{2} + 4}{4 x ^{2} + 2} : \frac{x}{4} + \frac{3}{4} + \frac{- x + 5}{2 ( 4 x ^{2} + 2 )}

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

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

\int \frac{x}{4} d x = \frac{x ^{2}}{8}

\int \frac{3}{4} d x = \frac{3}{4} x

\int \frac{- x + 5}{2 ( 4 x ^{2} + 2 )} d x = \frac{1}{2} (- \frac{1}{8} ln 4 x^{2} + 2 + \frac{5}{2 2} arctan (2 x))

= \frac{x ^{2}}{8} + \frac{3}{4} x + \frac{1}{2} (- \frac{1}{8} ln 4 x^{2} + 2 + \frac{5}{2 2} arctan (2 x))

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{8} + \frac{3}{4} x + \frac{1}{2} (- \frac{1}{8} ln 4 x^{2} + 2 + \frac{5}{2 2} arctan (2 x)) + C

Graph

Beliebte Beispiele

derivative f(x)=2x^4+3x+16

d er i v a t i v e f (x) = 2 x^{4} + 3 x + 16

d/(dt)(6/5-6/5 e^{-20t})

\frac{d}{d t} (\frac{6}{5} - \frac{6}{5} e^{- 20 t})

fläche y=x^3-6x^2+9x,y=x,x=0,x=2

a re a y = x^{3} - 6 x^{2} + 9 x, y = x, x = 0, x = 2

integral from 0 to infinity of 4xe^{-2x}

\int_{0}^{\infty} 4 x e^{- 2 x} d x

y^{''}+3y^'+2y=4t

y^{^{''}} + 3 y^{^{'}} + 2 y = 4 t