What is the integral of 3sec(2x)tan(2x) ?

The integral of 3sec(2x)tan(2x) is 3(1/2 cos(2x)+1/2 sin(2x)tan(2x))+C

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integral of 3sec(2x)tan(2x)

\int 3 sec (2 x) tan (2 x) d x

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

Solution steps

\int 3 sec (2 x) tan (2 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int sec (2 x) tan (2 x) d x

Rewrite using trig identities

= 3 \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)

= 3 (- \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)

= 3 (- (- \frac{1}{2} cos (2 x)) + \frac{1}{2} sin (2 x) tan (2 x))

Simplify

= 3 (\frac{1}{2} cos (2 x) + \frac{1}{2} sin (2 x) tan (2 x))

Add a constant to the solution

= 3 (\frac{1}{2} cos (2 x) + \frac{1}{2} sin (2 x) tan (2 x)) + C

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What is the integral of 3sec(2x)tan(2x) ?
The integral of 3sec(2x)tan(2x) is 3(1/2 cos(2x)+1/2 sin(2x)tan(2x))+C