Precálculo Ejemplos

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

Paso 1

Usa las propiedades de los logaritmos del producto, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Paso 2

Reescribe $\tan (x^{2})$ en términos de senos y cosenos.

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

Paso 3

Simplifica los términos.

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Paso 3.1

Aplica la propiedad distributiva.

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

Paso 3.2

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

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

Paso 3.3

Combina $\cos^{2} (x)$ y $\frac{\sin (x^{2})}{\cos (x^{2})}$ .

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

Paso 4

Simplifica cada término.

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Paso 4.1

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

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

Paso 4.2

Separa las fracciones.

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

Paso 4.3

Reescribe $\frac{\cos (x)}{\cos (x^{2})}$ como un producto.

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

Paso 4.4

Escribe $\cos (x)$ como una fracción con el denominador $1$ .

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

Paso 4.5

Simplifica.

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Paso 4.5.1

Divide $\cos (x)$ por $1$ .

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

Paso 4.5.2

Convierte de $\frac{1}{\cos (x^{2})}$ a $\sec (x^{2})$ .

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

Paso 4.6

Divide $\cos (x) \sin (x^{2})$ por $1$ .

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

Paso 4.7

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

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Paso 4.7.1

Eleva $\cos (x)$ a la potencia de $1$ .

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

Paso 4.7.2

Eleva $\cos (x)$ a la potencia de $1$ .

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

Paso 4.7.3

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 4.7.4

Suma $1$ y $1$ .

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