What is the integral of-csc^2((8x)/5) ?

The integral of-csc^2((8x)/5) is 5/8 cot((8x)/5)+C

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integral of-csc^2((8x)/5)

\int - csc^{2} (\frac{8 x}{5}) d x

\frac{5}{8} cot (\frac{8 x}{5}) + C

Solution steps

\int - csc^{2} (\frac{8 x}{5}) d x

Take the constant out:

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

= - \int csc^{2} (\frac{8 x}{5}) d x

Apply u-substitution

: \int csc^{2} (u) \frac{5}{8} d u

= - \int csc^{2} (u) \frac{5}{8} d u

Take the constant out:

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

= - \frac{5}{8} \cdot \int csc^{2} (u) d u

Use the common integral:

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

= - \frac{5}{8} (- cot (u))

Substitute back

u = \frac{8 x}{5}

= - \frac{5}{8} (- cot (\frac{8 x}{5}))

Simplify

= \frac{5}{8} cot (\frac{8 x}{5})

Add a constant to the solution

= \frac{5}{8} cot (\frac{8 x}{5}) + C

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What is the integral of-csc^2((8x)/5) ?
The integral of-csc^2((8x)/5) is 5/8 cot((8x)/5)+C