What is the integral of sec(x/8) ?

The integral of sec(x/8) is 8ln|tan(x/8)+sec(x/8)|+C

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integral of sec(x/8)

\int sec (\frac{x}{8}) d x

8 ln tan (\frac{x}{8}) + sec (\frac{x}{8}) + C

Solution steps

\int sec (\frac{x}{8}) d x

Apply u-substitution

= \int sec (u) \cdot 8 d u

Take the constant out:

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

= 8 \cdot \int sec (u) d u

Use the common integral:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= 8 ln ∣ tan (u) + sec (u) ∣

Substitute back

u = \frac{x}{8}

= 8 ln tan (\frac{x}{8}) + sec (\frac{x}{8})

Add a constant to the solution

= 8 ln tan (\frac{x}{8}) + sec (\frac{x}{8}) + C

What is the integral of sec(x/8) ?
The integral of sec(x/8) is 8ln|tan(x/8)+sec(x/8)|+C