What is the integral of 2tan(θ) ?

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

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integral of 2tan(θ)

\int 2 tan (θ) d θ

- 2 ln ∣ cos (θ) ∣ + C

Solution steps

\int 2 tan (θ) d θ

Take the constant out:

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

= 2 \cdot \int tan (θ) d θ

Rewrite using trig identities

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

Apply u-substitution

: \int - \frac{1}{u} d u

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 (- ln ∣ u ∣)

Substitute back

u = cos (θ)

= 2 (- ln ∣ cos (θ) ∣)

Simplify

= - 2 ln ∣ cos (θ) ∣

Add a constant to the solution

= - 2 ln ∣ cos (θ) ∣ + C

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