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

The integral of tan(x/5) is -5ln|cos(x/5)|+C

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integral of tan(x/5)

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

- 5 ln cos (\frac{x}{5}) + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

= \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 = cos (\frac{x}{5})

= - 5 ln cos (\frac{x}{5})

Add a constant to the solution

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

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