integrate sec^2(3x)(e^(tan3x))
To integrate the given expression, sec^2(3x)(e^(tan3x)), we can use a technique called substitution.
Let's start by letting u = tan(3x).
Taking the derivative of both sides with respect to x, we have:
du/dx = 3sec^2(3x).
Now, we can rewrite the expression to be integrated in terms of u:
sec^2(3x)(e^(tan3x)) = sec^2(3x)(e^u).
Next, we need to find du in terms of dx, so we can substitute it back into the original integral.
To do this, divide both sides of du/dx = 3sec^2(3x) by 3 and obtain: du/3 = sec^2(3x) dx.
Now, we have everything we need to perform the substitution.
Substituting u and du, the integral becomes:
∫ (sec^2(3x))(e^(tan3x)) dx = ∫ (e^u) du/3.
The integral on the right-hand side, ∫ (e^u) du/3, is now a simple integral to solve.
Integrating (e^u) with respect to u gives us e^u.
Thus, the result is:
∫ (sec^2(3x))(e^(tan3x)) dx = (1/3) e^u + C,
where C is the constant of integration.
Finally, substitute back u = tan(3x) into the result:
∫ (sec^2(3x))(e^(tan3x)) dx = (1/3) e^(tan(3x)) + C.
Therefore, the integral of sec^2(3x)(e^(tan3x)) with respect to x is (1/3) e^(tan(3x)) + C.