What is the integral of-2tan(2y) ?

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

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integral of-2tan(2y)

\int - 2 tan (2 y) d y

ln ∣ cos (2 y) ∣ + C

Solution steps

\int - 2 tan (2 y) d y

Take the constant out:

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

= - 2 \cdot \int tan (2 y) d y

Rewrite using trig identities

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

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 y)

= - 2 (- \frac{1}{2} ln ∣ cos (2 y) ∣)

Simplify

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

= ln ∣ cos (2 y) ∣

Add a constant to the solution

= ln ∣ cos (2 y) ∣ + C