What is the integral of csc(5t)cot(5t) ?

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

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integral of csc(5t)cot(5t)

\int csc (5 t) cot (5 t) d t

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

Solution steps

\int csc (5 t) cot (5 t) d t

Rewrite using trig identities

= \int \frac{cot ( 5 t )}{sin ( 5 t )} d t

Apply u-substitution

: \int \frac{cot ( u )}{5 sin ( u )} d u

= \int \frac{cot ( u )}{5 sin ( u )} 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 \frac{cot ( u )}{sin ( u )} d u

Rewrite using trig identities

= \frac{1}{5} \cdot \int \frac{cos ( u )}{sin ^{2} ( u )} d u

Apply u-substitution

: \int \frac{1}{v ^{2}} d v

= \frac{1}{5} \cdot \int \frac{1}{v ^{2}} d v

Apply the Power Rule

: - \frac{1}{v}

= \frac{1}{5} (- \frac{1}{v})

Substitute back

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

Simplify

\frac{1}{5} (- \frac{1}{sin ( 5 t )}) : - \frac{1}{5} csc (5 t)

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

Add a constant to the solution

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

\int \frac{5 x}{x ^{2} + 8} d x

x \to 4 lim (x + 5 - 3)

\frac{\partial}{\partial x} (e^{x - 2 y})

\int (\frac{1}{1 + cos ( x )}) d x

x \to 0 lim (x^{2} c x c x)

What is the integral of csc(5t)cot(5t) ?
The integral of csc(5t)cot(5t) is -1/5 csc(5t)+C