Calcolo Esempi

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

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

Passaggio 1

Applica la formula di riduzione.

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

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

Passaggio 2

L'integrale di

csc (x)

$\csc (x)$ rispetto a

x

$x$ è

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)

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

Passaggio 3

Semplifica la risposta.

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Passaggio 3.1

Riscrivi

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

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

- \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

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

Passaggio 3.2

Semplifica.

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Passaggio 3.2.1

cot (x)

$\cot (x)$ e

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Passaggio 3.2.2

csc (x)

$\csc (x)$ e

\frac{cot (x)}{2}

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

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

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

Passaggio 3.2.3

Per scrivere

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

$\frac{1}{2} \ln (| \csc (x) - \cot (x) |)$ come una frazione con un comune denominatore, moltiplicala per

\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

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

Passaggio 3.2.4

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

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

\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

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

Passaggio 3.2.5

Riduci i numeratori su un comune denominatore.

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

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

Passaggio 3.2.6

\frac{1}{2}

$\frac{1}{2}$ e

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

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

Passaggio 3.2.7

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

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

2

$2$ .

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

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

Passaggio 3.2.8

Sposta

2

$2$ alla sinistra di

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

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

Passaggio 3.2.9

Elimina il fattore comune di

2

$2$ .

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Passaggio 3.2.9.1

Elimina il fattore comune.

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

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

Passaggio 3.2.9.2

Dividi

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

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

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

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

Passaggio 3.3

Riordina i termini.

\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

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