What is the integral of sec^2(3x)+tan(6x) ?

The integral of sec^2(3x)+tan(6x) is 1/3 tan(3x)-1/6 ln|cos(6x)|+C

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integral of sec^2(3x)+tan(6x)

\int sec^{2} (3 x) + tan (6 x) d x

\frac{1}{3} tan (3 x) - \frac{1}{6} ln ∣ cos (6 x) ∣ + C

Solution steps

\int sec^{2} (3 x) + tan (6 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int sec^{2} (3 x) d x + \int tan (6 x) d x

\int sec^{2} (3 x) d x = \frac{1}{3} tan (3 x)

\int tan (6 x) d x = - \frac{1}{6} ln ∣ cos (6 x) ∣

= \frac{1}{3} tan (3 x) - \frac{1}{6} ln ∣ cos (6 x) ∣

Add a constant to the solution

= \frac{1}{3} tan (3 x) - \frac{1}{6} ln ∣ cos (6 x) ∣ + C

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What is the integral of sec^2(3x)+tan(6x) ?
The integral of sec^2(3x)+tan(6x) is 1/3 tan(3x)-1/6 ln|cos(6x)|+C