Final answer:
To evaluate the integral, use the substitution method by letting u = 3x + e^(3x) and v = 1 + e^(3x). Simplify and apply the power rule for integration to get the final result of (7/9) * (e^(3x + e^(3x))/(1 + e^(3x))) + C.
Step-by-step explanation:
To evaluate the integral, we can use the substitution method. Let u = 3x + e^(3x). Then differentiate u with respect to x to get du/dx = 3 + 3e^(3x). Rearranging the equation, we have dx = (1/(3 + 3e^(3x))) du. Substituting into the integral and simplifying, we get:
∫ 7e^(3x + e^(3x)) dx = ∫7e^u * (1/(3 + 3e^(3x))) du = (7/3) * ∫ e^u/(1 + e^(3x)) du.
We can further simplify the integral by letting v = 1 + e^(3x). Then differentiate v with respect to x to get dv/dx = 3e^(3x). Rearranging, we have dx = (1/(3e^(3x))) dv. Substituting into the integral and simplifying, we get:
(7/9) * ∫e^u/v * (1/v) dv = (7/9) * ∫e^u/v^2 dv.
To evaluate this integral, we can use the power rule for integration. The integral of e^u is e^u, and the integral of 1/v^2 is -1/v. Substituting back the values of u and v, we get the final result:
(7/9) * (e^(3x + e^(3x))/(1 + e^(3x))) + C