What is the integral of sec(x/9) ?

The integral of sec(x/9) is 9ln|tan(x/9)+sec(x/9)|+C

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integral of sec(x/9)

\int sec (\frac{x}{9}) d x

9 ln tan (\frac{x}{9}) + sec (\frac{x}{9}) + C

Solution steps

\int sec (\frac{x}{9}) d x

Apply u-substitution

: \int sec (u) \cdot 9 d u

= \int sec (u) \cdot 9 d u

Take the constant out:

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

= 9 \cdot \int sec (u) d u

Use the common integral:

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

= 9 ln ∣ tan (u) + sec (u) ∣

Substitute back

u = \frac{x}{9}

= 9 ln tan (\frac{x}{9}) + sec (\frac{x}{9})

Add a constant to the solution

= 9 ln tan (\frac{x}{9}) + sec (\frac{x}{9}) + C

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What is the integral of sec(x/9) ?
The integral of sec(x/9) is 9ln|tan(x/9)+sec(x/9)|+C