What is the integral of sec(ax) ?

The integral of sec(ax) is 1/a ln|tan(ax)+sec(ax)|+C

integral of sec(ax)

\int sec (a x) d x

\frac{1}{a} ln ∣ tan (a x) + sec (a x) ∣ + C

Solution steps

\int sec (a x) d x

Rewrite using trig identities

= \int \frac{1}{cos ( a x )} d x

Apply u-substitution

= \int \frac{1}{a cos ( u )} d u

Take the constant out:

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

= \frac{1}{a} \cdot \int \frac{1}{cos ( u )} d u

Rewrite using trig identities

= \frac{1}{a} \cdot \int sec (u) d u

Use the common integral:

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

= \frac{1}{a} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = a x

= \frac{1}{a} ln ∣ tan (a x) + sec (a x) ∣

Add a constant to the solution

= \frac{1}{a} ln ∣ tan (a x) + sec (a x) ∣ + C

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d er i v a t i v e (x - 1)^{3}

x \to 7 lim (\frac{\frac{7}{x} - 1}{x - 7})

What is the integral of sec(ax) ?
The integral of sec(ax) is 1/a ln|tan(ax)+sec(ax)|+C