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

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

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integral of 1/(tan(x/5))

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

5 ln sin (\frac{x}{5}) + C

Solution steps

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

Use the basic trigonometric identity:

\frac{1}{tan ( x )} = cot (x)

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

Rewrite using trig identities

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 ln ∣ u ∣

Substitute back

u = sin (\frac{x}{5})

= 5 ln sin (\frac{x}{5})

Add a constant to the solution

= 5 ln sin (\frac{x}{5}) + C

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