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Beliebt Rechnen >

integral of (4x)/((x^2+1)(x^2+2x+3))

Lösung

\int \frac{4 x}{( x ^{2} + 1 ) ( x ^{2} + 2 x + 3 )} d x

Lösung

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

Schritte zur Lösung

\int \frac{4 x}{( x ^{2} + 1 ) ( x ^{2} + 2 x + 3 )} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int \frac{x}{( x ^{2} + 1 ) ( x ^{2} + 2 x + 3 )} d x

Ermittle den Partialbruch von

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

= 4 \cdot \int \frac{x + 1}{4 ( x ^{2} + 1 )} + \frac{- x - 3}{4 ( x ^{2} + 2 x + 3 )} 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

= 4 (\int \frac{x + 1}{4 ( x ^{2} + 1 )} d x + \int \frac{- x - 3}{4 ( x ^{2} + 2 x + 3 )} d x)

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

\int \frac{- x - 3}{4 ( x ^{2} + 2 x + 3 )} d x = \frac{1}{8} (- ln x^{2} + 2 x + 3 - 22 arctan (\frac{1}{2} (x + 1)))

= 4 (\frac{1}{4} (\frac{1}{2} ln x^{2} + 1 + arctan (x)) + \frac{1}{8} (- ln x^{2} + 2 x + 3 - 22 arctan (\frac{1}{2} (x + 1))))

Vereinfache

4 (\frac{1}{4} (\frac{1}{2} ln x^{2} + 1 + arctan (x)) + \frac{1}{8} (- ln x^{2} + 2 x + 3 - 22 arctan (\frac{1}{2} (x + 1)))) : \frac{1}{2} ln x^{2} + 1 + arctan (x) - \frac{1}{2} ln x^{2} + 2 x + 3 - 2 arctan (\frac{1}{2} (x + 1))

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral from 0 to 10 of 20te^{-0.6t}

\int_{0}^{10} 20 t e^{- 0.6 t} d t

derivative of arccos(x/2)

\frac{d}{d x} (arccos (\frac{x}{2}))

y^'=((x^2+3y^2))/(2xy)

y^{^{'}} = \frac{( x ^{2} + 3 y ^{2} )}{2 x y}

limit as x approaches 3 of sqrt(x-13)

x \to 3 lim (x - 13)

y^{''}-y^'+y=13sin(t)

y^{^{''}} - y^{^{'}} + y = 13 sin (t)