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

The integral of-2tan(2x)sec(2x) is -cos(2x)-sin(2x)tan(2x)+C

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integral of-2tan(2x)sec(2x)

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

- cos (2 x) - sin (2 x) tan (2 x) + C

Solution steps

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

Rewrite using trig identities

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

Apply Integration By Parts

: - \int sin (2 x) d x + \frac{1}{2} sin (2 x) tan (2 x)

= - 2 (- \int sin (2 x) d x + \frac{1}{2} sin (2 x) tan (2 x))

\int sin (2 x) d x = - \frac{1}{2} cos (2 x)

= - 2 (- (- \frac{1}{2} cos (2 x)) + \frac{1}{2} sin (2 x) tan (2 x))

Simplify

- 2 (- (- \frac{1}{2} cos (2 x)) + \frac{1}{2} sin (2 x) tan (2 x)) : - cos (2 x) - sin (2 x) tan (2 x)

= - cos (2 x) - sin (2 x) tan (2 x)

Add a constant to the solution

= - cos (2 x) - sin (2 x) tan (2 x) + C

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What is the integral of-2tan(2x)sec(2x) ?
The integral of-2tan(2x)sec(2x) is -cos(2x)-sin(2x)tan(2x)+C