What is the integral of sec(t/3) ?

The integral of sec(t/3) is 3ln|tan(t/3)+sec(t/3)|+C

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integral of sec(t/3)

\int sec (\frac{t}{3}) d t

3 ln tan (\frac{t}{3}) + sec (\frac{t}{3}) + C

Solution steps

\int sec (\frac{t}{3}) d t

Apply u-substitution

: \int sec (u) \cdot 3 d u

= \int sec (u) \cdot 3 d u

Take the constant out:

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

= 3 \cdot \int sec (u) d u

Use the common integral:

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

= 3 ln ∣ tan (u) + sec (u) ∣

Substitute back

u = \frac{t}{3}

= 3 ln tan (\frac{t}{3}) + sec (\frac{t}{3})

Add a constant to the solution

= 3 ln tan (\frac{t}{3}) + sec (\frac{t}{3}) + C

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What is the integral of sec(t/3) ?
The integral of sec(t/3) is 3ln|tan(t/3)+sec(t/3)|+C