# int sec^2(3x) dx Find the indefinite integral

Indefinite integral are written in the form of int f(x) dx = F(x) +C

where: f(x) as the integrand

F(x) as the anti-derivative function

C  as the arbitrary constant known as constant of integration

For the given problem: int sec^2(3x)dx , the integrand function: f(x)=sec^2(3x) is in a form of trigonometric function.

To solve for the indefinite integral, we may apply the basic integration formula for secant function:

int sec^2(u)du=tan(u)+C

We may apply u-substitution when by letting:

u= 3x then du =3 dx or (du)/3 = dx .

Plug-in the values of u =3x and dx= (du)/3 , we get:

int sec^2(3x)dx =int sec^2(u)*(du)/3

=int (sec^2(u))/3du

Apply basic integration property: int c*f(x)dx= c int f(x)dx .

int (sec^2(u))/3du =(1/3)int sec^2(u)du

Then following the integral formula for secant, we get:

(1/3)int sec^2(u)du= 1/3tan(u)+C

Plug-in u =3x to solve for the indefinite integral F(x):

int sec^2(3x)dx=1/3tan(3x)+C

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