What is the integral of-e^{5x}csc^2(e^{5x}) ?

The integral of-e^{5x}csc^2(e^{5x}) is 1/5 cot(e^{5x})+C

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integral of-e^{5x}csc^2(e^{5x})

\int - e^{5 x} csc^{2} (e^{5 x}) d x

\frac{1}{5} cot (e^{5 x}) + C

Solution steps

\int - e^{5 x} csc^{2} (e^{5 x}) d x

Take the constant out:

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

= - \int e^{5 x} csc^{2} (e^{5 x}) d x

Apply u-substitution

: \int \frac{csc ^{2} ( u )}{5} d u

= - \int \frac{csc ^{2} ( u )}{5} d u

Take the constant out:

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

= - \frac{1}{5} \cdot \int csc^{2} (u) d u

Use the common integral:

\int csc^{2} (u) d u = - cot (u)

= - \frac{1}{5} (- cot (u))

Substitute back

u = e^{5 x}

= - \frac{1}{5} (- cot (e^{5 x}))

Simplify

= \frac{1}{5} cot (e^{5 x})

Add a constant to the solution

= \frac{1}{5} cot (e^{5 x}) + C

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What is the integral of-e^{5x}csc^2(e^{5x}) ?
The integral of-e^{5x}csc^2(e^{5x}) is 1/5 cot(e^{5x})+C