I need help with integration?

Question

Evaluate the integral
preferably using u substitution

"Integral symbol" sec^2 (3x-2) dx

thanks for the help

Answer 1

Integral ( sec^2(3x - 2) dx )

Note that sec^2(x) is one of our known derivatives. Once we get it to this form, we can easily integrate.

Let u = 3x - 2. Then
du = 3 dx, so
(1/3) du = dx.

Now, we substitute appropriately. Remember that this substitution includes dx.

Integral ( sec^2(u) (1/3) du )

Pulling out the constant (1/3) out of the integral,

(1/3) * Integral ( sec^2(u) du )

The integral of sec^2(u) is tan(u), so

(1/3) * tan(u) + C

Plugging in u = 3x - 2 back,

(1/3) * tan(3x - 2) + C

Answer 2

= 1/3 x (-(3x-2)-ctg(3x-2))