What is the integral of sec(2w)tan(2w) ?

The integral of sec(2w)tan(2w) is 1/2 cos(2w)+1/2 sin(2w)tan(2w)+C

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integral of sec(2w)tan(2w)

\int sec (2 w) tan (2 w) d w

\frac{1}{2} cos (2 w) + \frac{1}{2} sin (2 w) tan (2 w) + C

Solution steps

\int sec (2 w) tan (2 w) d w

Rewrite using trig identities

= \int \frac{sin ( 2 w )}{cos ^{2} ( 2 w )} d w

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

= \frac{1}{2} cos (2 w) + \frac{1}{2} sin (2 w) tan (2 w) + C

What is the integral of sec(2w)tan(2w) ?
The integral of sec(2w)tan(2w) is 1/2 cos(2w)+1/2 sin(2w)tan(2w)+C