Cálculo Ejemplos

\int \csc^{3} (x) d x

$\int \csc^{3} (x) d x$

Paso 1

Aplica la fórmula de reducción.

- \frac{\cot (x) \csc (x)}{2} + \frac{1}{2} \int \csc (x) d x

$- \frac{\cot (x) \csc (x)}{2} + \frac{1}{2} \int \csc (x) d x$

Paso 2

La integral de

\csc (x)

$\csc (x)$ con respecto a

x

$x$ es

\ln (| \csc (x) - \cot (x) |)

$\ln (| \csc (x) - \cot (x) |)$ .

- \frac{\cot (x) \csc (x)}{2} + \frac{1}{2} (\ln (| \csc (x) - \cot (x) |) + C)

$- \frac{\cot (x) \csc (x)}{2} + \frac{1}{2} (\ln (| \csc (x) - \cot (x) |) + C)$

Paso 3

Simplifica la respuesta.

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Paso 3.1

Reescribe

- \frac{\cot (x) \csc (x)}{2} + \frac{1}{2} (\ln (| \csc (x) - \cot (x) |) + C)

$- \frac{\cot (x) \csc (x)}{2} + \frac{1}{2} (\ln (| \csc (x) - \cot (x) |) + C)$ como

- \frac{1}{2} \cot (x) \csc (x) + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) + C

$- \frac{1}{2} \cot (x) \csc (x) + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) + C$ .

- \frac{1}{2} \cot (x) \csc (x) + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) + C

$- \frac{1}{2} \cot (x) \csc (x) + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) + C$

Paso 3.2

Simplifica.

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Paso 3.2.1

Combina

\cot (x)

$\cot (x)$ y

\frac{1}{2}

$\frac{1}{2}$ .

- \frac{\cot (x)}{2} \csc (x) + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) + C

$- \frac{\cot (x)}{2} \csc (x) + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) + C$

Paso 3.2.2

Combina

\csc (x)

$\csc (x)$ y

\frac{\cot (x)}{2}

$\frac{\cot (x)}{2}$ .

- \frac{\csc (x) \cot (x)}{2} + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) + C

$- \frac{\csc (x) \cot (x)}{2} + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) + C$

Paso 3.2.3

Para escribir

\frac{1}{2} \ln (| \csc (x) - \cot (x) |)

$\frac{1}{2} \ln (| \csc (x) - \cot (x) |)$ como una fracción con un denominador común, multiplica por

\frac{2}{2}

$\frac{2}{2}$ .

- \frac{\csc (x) \cot (x)}{2} + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) \cdot \frac{2}{2} + C

$- \frac{\csc (x) \cot (x)}{2} + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) \cdot \frac{2}{2} + C$

Paso 3.2.4

Combina

\frac{1}{2} \ln (| \csc (x) - \cot (x) |)

$\frac{1}{2} \ln (| \csc (x) - \cot (x) |)$ y

\frac{2}{2}

$\frac{2}{2}$ .

- \frac{\csc (x) \cot (x)}{2} + \frac{\frac{1}{2} \ln (| \csc (x) - \cot (x) |) \cdot 2}{2} + C

$- \frac{\csc (x) \cot (x)}{2} + \frac{\frac{1}{2} \ln (| \csc (x) - \cot (x) |) \cdot 2}{2} + C$

Paso 3.2.5

Combina los numeradores sobre el denominador común.

\frac{- \csc (x) \cot (x) + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) \cdot 2}{2} + C

$\frac{- \csc (x) \cot (x) + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) \cdot 2}{2} + C$

Paso 3.2.6

Combina

\frac{1}{2}

$\frac{1}{2}$ y

\ln (| \csc (x) - \cot (x) |)

$\ln (| \csc (x) - \cot (x) |)$ .

\frac{- \csc (x) \cot (x) + \frac{\ln (| \csc (x) - \cot (x) |)}{2} \cdot 2}{2} + C

$\frac{- \csc (x) \cot (x) + \frac{\ln (| \csc (x) - \cot (x) |)}{2} \cdot 2}{2} + C$

Paso 3.2.7

Combina

\frac{\ln (| \csc (x) - \cot (x) |)}{2}

$\frac{\ln (| \csc (x) - \cot (x) |)}{2}$ y

2

$2$ .

\frac{- \csc (x) \cot (x) + \frac{\ln (| \csc (x) - \cot (x) |) \cdot 2}{2}}{2} + C

$\frac{- \csc (x) \cot (x) + \frac{\ln (| \csc (x) - \cot (x) |) \cdot 2}{2}}{2} + C$

Paso 3.2.8

Mueve

2

$2$ a la izquierda de

\ln (| \csc (x) - \cot (x) |)

$\ln (| \csc (x) - \cot (x) |)$ .

\frac{- \csc (x) \cot (x) + \frac{2 \cdot \ln (| \csc (x) - \cot (x) |)}{2}}{2} + C

$\frac{- \csc (x) \cot (x) + \frac{2 \cdot \ln (| \csc (x) - \cot (x) |)}{2}}{2} + C$

Paso 3.2.9

Cancela el factor común de

2

$2$ .

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Paso 3.2.9.1

Cancela el factor común.

\frac{- \csc (x) \cot (x) + \frac{2 \ln (| \csc (x) - \cot (x) |)}{2}}{2} + C

$\frac{- \csc (x) \cot (x) + \frac{2 \ln (| \csc (x) - \cot (x) |)}{2}}{2} + C$

Paso 3.2.9.2

Divide

\ln (| \csc (x) - \cot (x) |)

$\ln (| \csc (x) - \cot (x) |)$ por

1

$1$ .

\frac{- \csc (x) \cot (x) + \ln (| \csc (x) - \cot (x) |)}{2} + C

$\frac{- \csc (x) \cot (x) + \ln (| \csc (x) - \cot (x) |)}{2} + C$

\frac{- \csc (x) \cot (x) + \ln (| \csc (x) - \cot (x) |)}{2} + C

$\frac{- \csc (x) \cot (x) + \ln (| \csc (x) - \cot (x) |)}{2} + C$

\frac{- \csc (x) \cot (x) + \ln (| \csc (x) - \cot (x) |)}{2} + C

$\frac{- \csc (x) \cot (x) + \ln (| \csc (x) - \cot (x) |)}{2} + C$

Paso 3.3

Reordena los términos.

\frac{1}{2} (- \csc (x) \cot (x) + \ln (| \csc (x) - \cot (x) |)) + C

$\frac{1}{2} (- \csc (x) \cot (x) + \ln (| \csc (x) - \cot (x) |)) + C$

\frac{1}{2} (- \csc (x) \cot (x) + \ln (| \csc (x) - \cot (x) |)) + C

$\frac{1}{2} (- \csc (x) \cot (x) + \ln (| \csc (x) - \cot (x) |)) + C$