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

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

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integral of sec(2z)tan(2z)

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

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

Solution steps

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

Rewrite using trig identities

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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