What is the integral of sec(5x) ?

The integral of sec(5x) is 1/5 ln|tan(5x)+sec(5x)|+C

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integral of sec(5x)

\int sec (5 x) d x

\frac{1}{5} ln ∣ tan (5 x) + sec (5 x) ∣ + C

Solution steps

\int sec (5 x) d x

Apply u-substitution

= \int sec (u) \frac{1}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int sec (u) d u

Use the common integral:

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

= \frac{1}{5} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = 5 x

= \frac{1}{5} ln ∣ tan (5 x) + sec (5 x) ∣

Add a constant to the solution

= \frac{1}{5} ln ∣ tan (5 x) + sec (5 x) ∣ + C

What is the integral of sec(5x) ?
The integral of sec(5x) is 1/5 ln|tan(5x)+sec(5x)|+C