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

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

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integral of sec(x/3)

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

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

Solution steps

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

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{x}{3}

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

Add a constant to the solution

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

(1 + x)^{\frac{1}{3}}

\int 5 x 4 - 3 x^{2} d x

\frac{u}{u ^{2} + 8}

\frac{d ^{2}}{d x ^{2}} (\frac{2}{3 x ^{4}})

\frac{d}{d x} (\frac{1}{y x ^{2}})

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