What is the integral of tan(2x-6) ?

The integral of tan(2x-6) is -1/2 ln|cos(2x-6)|+C

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integral of tan(2x-6)

\int tan (2 x - 6) d x

- \frac{1}{2} ln ∣ cos (2 x - 6) ∣ + C

Solution steps

\int tan (2 x - 6) d x

Rewrite using trig identities

= \int \frac{sin ( 2 x - 6 )}{cos ( 2 x - 6 )} d x

Apply u-substitution

: \int - \frac{1}{2 u} d u

= \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{2} ln ∣ u ∣

Substitute back

u = cos (2 x - 6)

= - \frac{1}{2} ln ∣ cos (2 x - 6) ∣

Add a constant to the solution

= - \frac{1}{2} ln ∣ cos (2 x - 6) ∣ + C

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What is the integral of tan(2x-6) ?
The integral of tan(2x-6) is -1/2 ln|cos(2x-6)|+C