What is the integral of sec((x+2)/3) ?

The integral of sec((x+2)/3) is 3ln|tan((x+2)/3)+sec((x+2)/3)|+C

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integral of sec((x+2)/3)

\int sec (\frac{x + 2}{3}) d x

3 ln tan (\frac{x + 2}{3}) + sec (\frac{x + 2}{3}) + C

Solution steps

\int sec (\frac{x + 2}{3}) d x

Apply u-substitution

: \int sec (\frac{u}{3}) d u

= \int sec (\frac{u}{3}) d u

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

= 3 ln ∣ tan (v) + sec (v) ∣

Substitute back

= 3 ln tan (\frac{x + 2}{3}) + sec (\frac{x + 2}{3})

Add a constant to the solution

= 3 ln tan (\frac{x + 2}{3}) + sec (\frac{x + 2}{3}) + C

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What is the integral of sec((x+2)/3) ?
The integral of sec((x+2)/3) is 3ln|tan((x+2)/3)+sec((x+2)/3)|+C