This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Set up the integral for the given expression. We need to integrate (3x) - e^3x with respect to x. ((3x) - e^3x)\,dx Step 2: Integrate each term separately. The integral of a difference is the difference of the integrals. (3x)\,dx - e^3x\,dx For the first term, (ax)\,dx = -(1)/(a)(ax) + C. Here a=3. (3x)\,dx = -(1)/(3)(3x) For the second term, e^ax\,dx = (1)/(a)e^ax + C. Here a=3. e^3x\,dx = (1)/(3)e^3x Step 3: Combine the results and add the constant of integration. -(1)/(3)(3x) - (1)/(3)e^3x + C The final answer is -(1)/(3)(3x) - (1)/(3)e^3x + C.