What is the integral of-6e^{-4csc(3x)}cot(3x)csc(3x) ?

The integral of-6e^{-4csc(3x)}cot(3x)csc(3x) is -1/2 e^{-4csc(3x)}+C

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integral of-6e^{-4csc(3x)}cot(3x)csc(3x)

\int - 6 e^{- 4 c s c (3 x)} cot (3 x) csc (3 x) d x

- \frac{1}{2} e^{- 4 c s c (3 x)} + C

Solution steps

\int - 6 e^{- 4 c s c (3 x)} cot (3 x) csc (3 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 6 \cdot \int e^{- 4 c s c (3 x)} cot (3 x) csc (3 x) d x

Apply u-substitution

= - 6 \cdot \int \frac{e ^{u}}{12} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 6 \cdot \frac{1}{12} \cdot \int e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= - 6 \cdot \frac{1}{12} e^{u}

Substitute back

u = - 4 csc (3 x)

= - 6 \cdot \frac{1}{12} e^{- 4 c s c (3 x)}

Simplify

- 6 \cdot \frac{1}{12} e^{- 4 c s c (3 x)} : - \frac{1}{2} e^{- 4 c s c (3 x)}

= - \frac{1}{2} e^{- 4 c s c (3 x)}

Add a constant to the solution

= - \frac{1}{2} e^{- 4 c s c (3 x)} + C

What is the integral of-6e^{-4csc(3x)}cot(3x)csc(3x) ?
The integral of-6e^{-4csc(3x)}cot(3x)csc(3x) is -1/2 e^{-4csc(3x)}+C