What is the integral of sec((4x)/3) ?

The integral of sec((4x)/3) is 3/4 ln|tan(4/3 x)+sec(4/3 x)|+C

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integral of sec((4x)/3)

\int sec (\frac{4 x}{3}) d x

\frac{3}{4} ln tan (\frac{4}{3} x) + sec (\frac{4}{3} x) + C

Solution steps

\int sec (\frac{4 x}{3}) d x

\frac{4 x}{3} = \frac{4}{3} x

= \int sec (\frac{4}{3} x) d x

Apply u-substitution

= \int sec (u) \frac{3}{4} d u

Take the constant out:

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

= \frac{3}{4} \cdot \int sec (u) d u

Use the common integral:

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

= \frac{3}{4} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = \frac{4}{3} x

= \frac{3}{4} ln tan (\frac{4}{3} x) + sec (\frac{4}{3} x)

Add a constant to the solution

= \frac{3}{4} ln tan (\frac{4}{3} x) + sec (\frac{4}{3} x) + C

What is the integral of sec((4x)/3) ?
The integral of sec((4x)/3) is 3/4 ln|tan(4/3 x)+sec(4/3 x)|+C