What is the integral of 1/(tan(5t)) ?

The integral of 1/(tan(5t)) is 1/5 ln|sin(5t)|+C

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integral of 1/(tan(5t))

\int \frac{1}{tan ( 5 t )} d t

\frac{1}{5} ln ∣ sin (5 t) ∣ + C

Solution steps

\int \frac{1}{tan ( 5 t )} d t

Rewrite using trig identities

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

Apply u-substitution

: \int \frac{1}{5 u} d u

= \int \frac{1}{5 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{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{5} ln ∣ u ∣

Substitute back

u = sin (5 t)

= \frac{1}{5} ln ∣ sin (5 t) ∣

Add a constant to the solution

= \frac{1}{5} ln ∣ sin (5 t) ∣ + C

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What is the integral of 1/(tan(5t)) ?
The integral of 1/(tan(5t)) is 1/5 ln|sin(5t)|+C