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

Lösung

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

\frac{3}{4 23} (23 ln \frac{32}{23} + \frac{16 x ^{2} + 24 x}{23} - 6 arctan (\frac{4 x + 3}{23})) + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Vervollständige das Quadrat

2 x^{2} + 3 x + 4 : 2 (x + \frac{3}{4})^{2} + \frac{23}{8}

= 3 \cdot \int \frac{x}{2 ( x + \frac{3}{4} ) ^{2} + \frac{23}{8}} d x

Wende U-Substitution an

= 3 \cdot \int \frac{2 ( 4 u - 3 )}{16 u ^{2} + 23} d u

Entferne die Konstante:

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

= 3 \cdot 2 \cdot \int \frac{4 u - 3}{16 u ^{2} + 23} d u

Wende integrale Substitution an

= 3 \cdot 2 \cdot \int \frac{23 v - 3}{4 23 ( v ^{2} + 1 )} d v

Entferne die Konstante:

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

= 3 \cdot 2 \cdot \frac{1}{4 23} \cdot \int \frac{23 v - 3}{v ^{2} + 1} d v

Multipliziere aus

\frac{23 v - 3}{v ^{2} + 1} : \frac{23 v}{v ^{2} + 1} - \frac{3}{v ^{2} + 1}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 \cdot 2 \cdot \frac{1}{4 23} (\int \frac{23 v}{v ^{2} + 1} d v - \int \frac{3}{v ^{2} + 1} d v)

\int \frac{23 v}{v ^{2} + 1} d v = \frac{23}{2} ln v^{2} + 1

\int \frac{3}{v ^{2} + 1} d v = 3 arctan (v)

= 3 \cdot 2 \cdot \frac{1}{4 23} (\frac{23}{2} ln v^{2} + 1 - 3 arctan (v))

Ersetze zurück

= 3 \cdot 2 \cdot \frac{1}{4 23} (\frac{23}{2} ln (\frac{4}{23} (x + \frac{3}{4}))^{2} + 1 - 3 arctan (\frac{4}{23} (x + \frac{3}{4})))

Vereinfache

3 \cdot 2 \cdot \frac{1}{4 23} (\frac{23}{2} ln (\frac{4}{23} (x + \frac{3}{4}))^{2} + 1 - 3 arctan (\frac{4}{23} (x + \frac{3}{4}))) : \frac{3}{4 23} (23 ln \frac{32}{23} + \frac{16 x ^{2} + 24 x}{23} - 6 arctan (\frac{4 x + 3}{23}))

= \frac{3}{4 23} (23 ln \frac{32}{23} + \frac{16 x ^{2} + 24 x}{23} - 6 arctan (\frac{4 x + 3}{23}))

Füge eine Konstante zur Lösung hinzu

= \frac{3}{4 23} (23 ln \frac{32}{23} + \frac{16 x ^{2} + 24 x}{23} - 6 arctan (\frac{4 x + 3}{23})) + C

\frac{\partial}{\partial x} (2 x^{6 y})

\int e x sin (x) d x

s l o p e \frac{x}{2}

\frac{d}{d x} (C e^{2 x})

(4 + t^{2}) (\frac{d y}{d t}) + 2 t y = 4 t