What is the integral of 2sec(2x) ?

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

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integral of 2sec(2x)

\int 2 sec (2 x) d x

ln ∣ tan (2 x) + sec (2 x) ∣ + C

Solution steps

\int 2 sec (2 x) d x

Take the constant out:

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

= 2 \cdot \int sec (2 x) d x

Apply u-substitution

= 2 \cdot \int sec (u) \frac{1}{2} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int sec (u) d u

Use the common integral:

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

= 2 \cdot \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = 2 x

= 2 \cdot \frac{1}{2} ln ∣ tan (2 x) + sec (2 x) ∣

Simplify

2 \cdot \frac{1}{2} ln ∣ tan (2 x) + sec (2 x) ∣ : ln ∣ tan (2 x) + sec (2 x) ∣

= ln ∣ tan (2 x) + sec (2 x) ∣

Add a constant to the solution

= ln ∣ tan (2 x) + sec (2 x) ∣ + C

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