What is the integral of sec(x/6) ?

The integral of sec(x/6) is 6ln|tan(x/6)+sec(x/6)|+C

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integral of sec(x/6)

\int sec (\frac{x}{6}) d x

6 ln tan (\frac{x}{6}) + sec (\frac{x}{6}) + C

Solution steps

\int sec (\frac{x}{6}) d x

Apply u-substitution

: \int sec (u) \cdot 6 d u

= \int sec (u) \cdot 6 d u

Take the constant out:

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

= 6 \cdot \int sec (u) d u

Use the common integral:

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

= 6 ln ∣ tan (u) + sec (u) ∣

Substitute back

u = \frac{x}{6}

= 6 ln tan (\frac{x}{6}) + sec (\frac{x}{6})

Add a constant to the solution

= 6 ln tan (\frac{x}{6}) + sec (\frac{x}{6}) + C

(- 4, 5), (5, - 22)

x \to 6 lim (π)

\int \frac{16 x ^{3} - 32 x ^{2} + 5}{x ^{2} - 2 x} d x

e^{- 4 x}

\int 8^{3 x + 5} d x

What is the integral of sec(x/6) ?
The integral of sec(x/6) is 6ln|tan(x/6)+sec(x/6)|+C