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.
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To evaluate the integral e^3x 5x \,dx, we will use integration by parts twice. The formula for integration by parts is u \,dv = uv - v \,du. Let I = e^3x 5x \,dx. Step 1: Apply integration by parts for the first time. Let u = 5x and dv = e^3x \,dx. Then, we find du and v: du = -5 5x \,dx v = e^3x \,dx = (1)/(3) e^3x Substitute these into the integration by parts formula: I = ( 5x) ((1)/(3) e^3x) - ((1)/(3) e^3x) (-5 5x) \,dx I = (1)/(3) e^3x 5x + (5)/(3) e^3x 5x \,dx Step 2: Apply integration by parts for the second time to the new integral e^3x 5x \,dx. Let I_1 = e^3x 5x \,dx. Let u = 5x and dv = e^3x \,dx. Then, we find du and v: du = 5 5x \,dx v = e^3x \,dx = (1)/(3) e^3x Substitute these into the integration by parts formula: I_1 = ( 5x) ((1)/(3) e^3x) - ((1)/(3) e^3x) (5 5x) \,dx I_1 = (1)/(3) e^3x 5x - (5)/(3) e^3x 5x \,dx Notice that the integral on the right side is our original integral I. So, we have: I_1 = (1)/(3) e^3x 5x - (5)/(3) I Step 3: Substitute I_1 back into the equation for I from Step 1. I = (1)/(3) e^3x 5x + (5)/(3) ( (1)/(3) e^3x 5x - (5)/(3) I ) I = (1)/(3) e^3x 5x + (5)/(9) e^3x 5x - (25)/(9) I Step 4: Solve the equation for I. Add (25)/(9) I to both sides: I + (25)/(9) I = (1)/(3) e^3x 5x + (5)/(9) e^3x 5x (1 + (25)/(9)) I = (1)/(3) e^3x 5x + (5)/(9) e^3x 5x ((9+25)/(9)) I = (1)/(3) e^3x 5x + (5)/(9) e^3x 5x (34)/(9) I = (1)/(3) e^3x 5x + (5)/(9) e^3x 5x Multiply both sides by (9)/(34): I = (9)/(34) ( (1)/(3) e^3x 5x + (5)/(9) e^3x 5x ) I = (9)/(34) · (1)/(3) e^3x 5x + (9)/(34) · (5)/(9) e^3x 5x I = (3)/(34) e^3x 5x + (5)/(34) e^3x 5x Step 5: Add the constant of integration. I = e^3x34 (3 5x + 5 5x) + C The final answer is e^3x34 (3 5x + 5 5x) + C.