Cálculo Exemplos

y = \frac{x \sin (x)}{1 + \cos (x)}

$y = \frac{x \sin (x)}{1 + \cos (x)}$

Etapa 1

Diferencie usando a regra do quociente, que determina que

\frac{d}{d x} [\frac{f (x)}{g (x)}]

$\frac{d}{d x} [\frac{f (x)}{g (x)}]$ é

\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}

$\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ , em que

f (x) = x \sin (x)

$f (x) = x \sin (x)$ e

g (x) = 1 + \cos (x)

$g (x) = 1 + \cos (x)$ .

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

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

Etapa 2

Diferencie usando a regra do produto, que determina que

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ é

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ , em que

f (x) = x

$f (x) = x$ e

g (x) = \sin (x)

$g (x) = \sin (x)$ .

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

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

Etapa 3

A derivada de

\sin (x)

$\sin (x)$ em relação a

x

$x$ é

\cos (x)

$\cos (x)$ .

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

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

Etapa 4

Diferencie.

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Etapa 4.1

Diferencie usando a regra da multiplicação de potências, que determina que

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ é

n x^{n - 1}

$n x^{n - 1}$ , em que

n = 1

$n = 1$ .

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

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

Etapa 4.2

Multiplique

\sin (x)

$\sin (x)$ por

1

$1$ .

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

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

Etapa 4.3

De acordo com a regra da soma, a derivada de

1 + \cos (x)

$1 + \cos (x)$ com relação a

x

$x$ é

\frac{d}{d x} [1] + \frac{d}{d x} [\cos (x)]

$\frac{d}{d x} [1] + \frac{d}{d x} [\cos (x)]$ .

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

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

Etapa 4.4

Como

1

$1$ é constante em relação a

x

$x$ , a derivada de

1

$1$ em relação a

x

$x$ é

0

$0$ .

\frac{(1 + \cos (x)) (x \cos (x) + \sin (x)) - x \sin (x) (0 + \frac{d}{d x} [\cos (x)])}{{(1 + \cos (x))}^{2}}

$\frac{(1 + \cos (x)) (x \cos (x) + \sin (x)) - x \sin (x) (0 + \frac{d}{d x} [\cos (x)])}{{(1 + \cos (x))}^{2}}$

Etapa 4.5

Some

0

$0$ e

\frac{d}{d x} [\cos (x)]

$\frac{d}{d x} [\cos (x)]$ .

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

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

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

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

Etapa 5

A derivada de

\cos (x)

$\cos (x)$ em relação a

x

$x$ é

- \sin (x)

$- \sin (x)$ .

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

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

Etapa 6

Multiplique.

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Etapa 6.1

Multiplique

- 1

$- 1$ por

- 1

$- 1$ .

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

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

Etapa 6.2

Multiplique

x

$x$ por

1

$1$ .

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

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

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

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

Etapa 7

Eleve

\sin (x)

$\sin (x)$ à potência de

1

$1$ .

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

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

Etapa 8

Eleve

\sin (x)

$\sin (x)$ à potência de

1

$1$ .

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

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

Etapa 9

Use a regra da multiplicação de potências

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

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

Etapa 10

Some

1

$1$ e

1

$1$ .

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

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

Etapa 11

Simplifique.

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Etapa 11.1

Simplifique o numerador.

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Etapa 11.1.1

Simplifique cada termo.

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Etapa 11.1.1.1

Expanda

(1 + \cos (x)) (x \cos (x) + \sin (x))

$(1 + \cos (x)) (x \cos (x) + \sin (x))$ usando o método FOIL.

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Etapa 11.1.1.1.1

Aplique a propriedade distributiva.

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

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

Etapa 11.1.1.1.2

Aplique a propriedade distributiva.

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

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

Etapa 11.1.1.1.3

Aplique a propriedade distributiva.

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

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

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

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

Etapa 11.1.1.2

Simplifique cada termo.

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Etapa 11.1.1.2.1

Multiplique

x \cos (x)

$x \cos (x)$ por

1

$1$ .

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

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

Etapa 11.1.1.2.2

Multiplique

\sin (x)

$\sin (x)$ por

1

$1$ .

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

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

Etapa 11.1.1.2.3

Multiplique

\cos (x) (x \cos (x))

$\cos (x) (x \cos (x))$ .

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Etapa 11.1.1.2.3.1

Eleve

\cos (x)

$\cos (x)$ à potência de

1

$1$ .

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

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

Etapa 11.1.1.2.3.2

Eleve

\cos (x)

$\cos (x)$ à potência de

1

$1$ .

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

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

Etapa 11.1.1.2.3.3

Use a regra da multiplicação de potências

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

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

Etapa 11.1.1.2.3.4

Some

1

$1$ e

1

$1$ .

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

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

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

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

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

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

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

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

Etapa 11.1.2

Mova

x \sin^{2} (x)

$x \sin^{2} (x)$ .

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

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

Etapa 11.1.3

Fatore

x

$x$ de

x \cos^{2} (x)

$x \cos^{2} (x)$ .

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

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

Etapa 11.1.4

Fatore

x

$x$ de

x \sin^{2} (x)

$x \sin^{2} (x)$ .

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

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

Etapa 11.1.5

Fatore

x

$x$ de

x (\cos^{2} (x)) + x (\sin^{2} (x))

$x (\cos^{2} (x)) + x (\sin^{2} (x))$ .

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

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

Etapa 11.1.6

Reorganize os termos.

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

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

Etapa 11.1.7

Aplique a identidade trigonométrica fundamental.

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

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

Etapa 11.1.8

Multiplique

x

$x$ por

1

$1$ .

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

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

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

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

Etapa 11.2

Reordene os termos.

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

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

Etapa 11.3

Simplifique o numerador.

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Etapa 11.3.1

Fatore o máximo divisor comum de cada grupo.

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Etapa 11.3.1.1

Agrupe os dois primeiros termos e os dois últimos termos.

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

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

Etapa 11.3.1.2

Fatore o máximo divisor comum (MDC) de cada grupo.

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

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

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

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

Etapa 11.3.2

Fatore o polinômio desmembrando o máximo divisor comum,

x + \sin (x)

$x + \sin (x)$ .

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

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

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

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

Etapa 11.4

Cancele o fator comum de

\cos (x) + 1

$\cos (x) + 1$ e

{(1 + \cos (x))}^{2}

${(1 + \cos (x))}^{2}$ .

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Etapa 11.4.1

Reordene os termos.

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

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

Etapa 11.4.2

Fatore

1 + \cos (x)

$1 + \cos (x)$ de

(x + \sin (x)) (1 + \cos (x))

$(x + \sin (x)) (1 + \cos (x))$ .

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

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

Etapa 11.4.3

Cancele os fatores comuns.

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Etapa 11.4.3.1

Fatore

1 + \cos (x)

$1 + \cos (x)$ de

{(1 + \cos (x))}^{2}

${(1 + \cos (x))}^{2}$ .

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

$\frac{(1 + \cos (x)) (x + \sin (x))}{(1 + \cos (x)) (1 + \cos (x))}$

Etapa 11.4.3.2

Cancele o fator comum.

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

$\frac{(1 + \cos (x)) (x + \sin (x))}{(1 + \cos (x)) (1 + \cos (x))}$

Etapa 11.4.3.3

Reescreva a expressão.

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

$\frac{x + \sin (x)}{1 + \cos (x)}$

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

$\frac{x + \sin (x)}{1 + \cos (x)}$

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

$\frac{x + \sin (x)}{1 + \cos (x)}$

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

$\frac{x + \sin (x)}{1 + \cos (x)}$