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

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

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integral of 2tan(2x)

\int 2 tan (2 x) d x

- ln ∣ cos (2 x) ∣ + C

Solution steps

\int 2 tan (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 tan (2 x) d x

Rewrite using trig identities

= 2 \cdot \int \frac{sin ( 2 x )}{cos ( 2 x )} d x

Apply u-substitution

= 2 \cdot \int - \frac{1}{2 u} d u

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 (- \frac{1}{2} ln ∣ u ∣)

Substitute back

u = cos (2 x)

= 2 (- \frac{1}{2} ln ∣ cos (2 x) ∣)

Simplify

2 (- \frac{1}{2} ln ∣ cos (2 x) ∣) : - ln ∣ cos (2 x) ∣

= - ln ∣ cos (2 x) ∣

Add a constant to the solution

= - ln ∣ cos (2 x) ∣ + C