What is the integral of csc^2(5t) ?

The integral of csc^2(5t) is -1/5 cot(5t)+C

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integral of csc^2(5t)

\int csc^{2} (5 t) d t

- \frac{1}{5} cot (5 t) + C

Solution steps

\int csc^{2} (5 t) d t

Apply u-substitution

: \int csc^{2} (u) \frac{1}{5} d u

= \int csc^{2} (u) \frac{1}{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 = 5 t

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

Simplify

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

Add a constant to the solution

= - \frac{1}{5} cot (5 t) + C

y = \frac{3}{x}, y = 12 x, y = \frac{x}{3}

y^{^{'}} + 5 y = e^{(- 2 x)}

p (θ) = 3 θ

y = (x + 1)^{x}

\int_{0}^{2} 2 π x (2 x^{2} - x^{3}) d x