What is the integral of 12sec(3x)tan(3x) ?

The integral of 12sec(3x)tan(3x) is 4sec(3x)+C

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integral of 12sec(3x)tan(3x)

\int 12 sec (3 x) tan (3 x) d x

4 sec (3 x) + C

Solution steps

\int 12 sec (3 x) tan (3 x) d x

Take the constant out:

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

= 12 \cdot \int sec (3 x) tan (3 x) d x

Apply u-substitution

: \int \frac{1}{3} d u

= 12 \cdot \int \frac{1}{3} d u

Integral of a constant:

\int a d x = a x

= 12 \cdot \frac{1}{3} u

Substitute back

u = sec (3 x)

= 12 \cdot \frac{1}{3} sec (3 x)

Simplify

12 \cdot \frac{1}{3} sec (3 x) : 4 sec (3 x)

= 4 sec (3 x)

Add a constant to the solution

= 4 sec (3 x) + C

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What is the integral of 12sec(3x)tan(3x) ?
The integral of 12sec(3x)tan(3x) is 4sec(3x)+C