What is the integral of 4/5 csc^2(x/5) ?

The integral of 4/5 csc^2(x/5) is -4cot(x/5)+C

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integral of 4/5 csc^2(x/5)

\int \frac{4}{5} csc^{2} (\frac{x}{5}) d x

- 4 cot (\frac{x}{5}) + C

Solution steps

\int \frac{4}{5} csc^{2} (\frac{x}{5}) d x

Take the constant out:

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

= \frac{4}{5} \cdot \int csc^{2} (\frac{x}{5}) d x

Rewrite using trig identities

= \frac{4}{5} \cdot \int 1 + cot^{2} (\frac{x}{5}) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{4}{5} (\int 1 d x + \int cot^{2} (\frac{x}{5}) d x)

\int 1 d x = x

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

= \frac{4}{5} (x - x - 5 cot (\frac{x}{5}))

Simplify

\frac{4}{5} (x - x - 5 cot (\frac{x}{5})) : - 4 cot (\frac{x}{5})

= - 4 cot (\frac{x}{5})

Add a constant to the solution

= - 4 cot (\frac{x}{5}) + C

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What is the integral of 4/5 csc^2(x/5) ?
The integral of 4/5 csc^2(x/5) is -4cot(x/5)+C