Analysis Beispiele

\frac{d x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)}

$\frac{d x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)}$

Schritt 1

\frac{x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)}

$\frac{x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)}$ konstant bezüglich

d

$d$ ist, ziehe

\frac{x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)}

$\frac{x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)}$ aus dem Integral.

\frac{x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)} \int d d \cdot d

$\frac{x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)} \int d d \cdot d$

Schritt 2

Gemäß der Potenzregel ist das Integral von

d

$d$ nach

d

$d$ gleich

\frac{1}{2} d^{2}

$\frac{1}{2} d^{2}$ .

\frac{x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)} (\frac{1}{2} d^{2} + C)

$\frac{x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)} (\frac{1}{2} d^{2} + C)$

Schritt 3

Vereinfache die Lösung.

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Schritt 3.1

Schreibe

\frac{x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)} (\frac{1}{2} d^{2} + C)

$\frac{x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)} (\frac{1}{2} d^{2} + C)$ als

\frac{x}{2 + \cos (x) - 2 \sin (x)} \csc (x) (\frac{1}{2} d^{2}) + C

$\frac{x}{2 + \cos (x) - 2 \sin (x)} \csc (x) (\frac{1}{2} d^{2}) + C$ um.

\frac{x}{2 + \cos (x) - 2 \sin (x)} \csc (x) (\frac{1}{2} d^{2}) + C

$\frac{x}{2 + \cos (x) - 2 \sin (x)} \csc (x) (\frac{1}{2} d^{2}) + C$

Schritt 3.2

Vereinfache.

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Schritt 3.2.1

Kombiniere

\frac{x}{2 + \cos (x) - 2 \sin (x)}

$\frac{x}{2 + \cos (x) - 2 \sin (x)}$ und

\csc (x)

$\csc (x)$ .

\frac{x \csc (x)}{2 + \cos (x) - 2 \sin (x)} (\frac{1}{2} d^{2}) + C

$\frac{x \csc (x)}{2 + \cos (x) - 2 \sin (x)} (\frac{1}{2} d^{2}) + C$

Schritt 3.2.2

Kombiniere

\frac{1}{2}

$\frac{1}{2}$ und

d^{2}

$d^{2}$ .

\frac{x \csc (x)}{2 + \cos (x) - 2 \sin (x)} \cdot \frac{d^{2}}{2} + C

$\frac{x \csc (x)}{2 + \cos (x) - 2 \sin (x)} \cdot \frac{d^{2}}{2} + C$

Schritt 3.2.3

Mutltipliziere

\frac{x \csc (x)}{2 + \cos (x) - 2 \sin (x)}

$\frac{x \csc (x)}{2 + \cos (x) - 2 \sin (x)}$ mit

\frac{d^{2}}{2}

$\frac{d^{2}}{2}$ .

\frac{x \csc (x) d^{2}}{(2 + \cos (x) - 2 \sin (x)) \cdot 2} + C

$\frac{x \csc (x) d^{2}}{(2 + \cos (x) - 2 \sin (x)) \cdot 2} + C$

Schritt 3.2.4

Bringe

2

$2$ auf die linke Seite von

2 + \cos (x) - 2 \sin (x)

$2 + \cos (x) - 2 \sin (x)$ .

\frac{x \csc (x) d^{2}}{2 (2 + \cos (x) - 2 \sin (x))} + C

$\frac{x \csc (x) d^{2}}{2 (2 + \cos (x) - 2 \sin (x))} + C$

\frac{x \csc (x) d^{2}}{2 (2 + \cos (x) - 2 \sin (x))} + C

$\frac{x \csc (x) d^{2}}{2 (2 + \cos (x) - 2 \sin (x))} + C$

Schritt 3.3

Vereinfache.

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Schritt 3.3.1

Vereinfache den Zähler.

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Schritt 3.3.1.1

Schreibe

\csc (x)

$\csc (x)$ mithilfe von Sinus und Kosinus um.

\frac{x \frac{1}{\sin (x)} d^{2}}{2 (2 + \cos (x) - 2 \sin (x))} + C

$\frac{x \frac{1}{\sin (x)} d^{2}}{2 (2 + \cos (x) - 2 \sin (x))} + C$

Schritt 3.3.1.2

Kombiniere Exponenten.

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Schritt 3.3.1.2.1

Kombiniere

x

$x$ und

\frac{1}{\sin (x)}

$\frac{1}{\sin (x)}$ .

\frac{\frac{x}{\sin (x)} d^{2}}{2 (2 + \cos (x) - 2 \sin (x))} + C

$\frac{\frac{x}{\sin (x)} d^{2}}{2 (2 + \cos (x) - 2 \sin (x))} + C$

Schritt 3.3.1.2.2

Kombiniere

\frac{x}{\sin (x)}

$\frac{x}{\sin (x)}$ und

d^{2}

$d^{2}$ .

\frac{\frac{x d^{2}}{\sin (x)}}{2 (2 + \cos (x) - 2 \sin (x))} + C

$\frac{\frac{x d^{2}}{\sin (x)}}{2 (2 + \cos (x) - 2 \sin (x))} + C$

\frac{\frac{x d^{2}}{\sin (x)}}{2 (2 + \cos (x) - 2 \sin (x))} + C

$\frac{\frac{x d^{2}}{\sin (x)}}{2 (2 + \cos (x) - 2 \sin (x))} + C$

\frac{\frac{x d^{2}}{\sin (x)}}{2 (2 + \cos (x) - 2 \sin (x))} + C

$\frac{\frac{x d^{2}}{\sin (x)}}{2 (2 + \cos (x) - 2 \sin (x))} + C$

Schritt 3.3.2

Multipliziere den Zähler mit dem Kehrwert des Nenners.

\frac{x d^{2}}{\sin (x)} \cdot \frac{1}{2 (2 + \cos (x) - 2 \sin (x))} + C

$\frac{x d^{2}}{\sin (x)} \cdot \frac{1}{2 (2 + \cos (x) - 2 \sin (x))} + C$

Schritt 3.3.3

Kombinieren.

\frac{x d^{2} \cdot 1}{\sin (x) (2 (2 + \cos (x) - 2 \sin (x)))} + C

$\frac{x d^{2} \cdot 1}{\sin (x) (2 (2 + \cos (x) - 2 \sin (x)))} + C$

Schritt 3.3.4

Mutltipliziere

x

$x$ mit

1

$1$ .

\frac{d^{2} \cdot x}{\sin (x) \cdot 2 (2 + \cos (x) - 2 \sin (x))} + C

$\frac{d^{2} \cdot x}{\sin (x) \cdot 2 (2 + \cos (x) - 2 \sin (x))} + C$

Schritt 3.3.5

Bringe

2

$2$ auf die linke Seite von

\sin (x)

$\sin (x)$ .

\frac{d^{2} x}{2 \sin (x) (2 + \cos (x) - 2 \sin (x))} + C

$\frac{d^{2} x}{2 \sin (x) (2 + \cos (x) - 2 \sin (x))} + C$

\frac{d^{2} x}{2 \sin (x) (2 + \cos (x) - 2 \sin (x))} + C

$\frac{d^{2} x}{2 \sin (x) (2 + \cos (x) - 2 \sin (x))} + C$

\frac{d^{2} x}{2 \sin (x) (2 + \cos (x) - 2 \sin (x))} + C

$\frac{d^{2} x}{2 \sin (x) (2 + \cos (x) - 2 \sin (x))} + C$