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

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

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integral of sec(2y)tan(2y)

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

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

Solution steps

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

Rewrite using trig identities

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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