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
Right ThegirlGodishelping🥰💙🫠, let's go. We need to evaluate the integral: e^x x \, dx This integral requires integration by parts twice. The formula for integration by parts is u \, dv = uv - v \, du. Step 1: Apply integration by parts for the first time. Let I = e^x x \, dx. Choose u = x and dv = e^x \, dx. Then, du = x \, dx and v = e^x. Applying the formula: I = e^x x - e^x x \, dx (*) Step 2: Apply integration by parts again to the new integral e^x x \, dx. Choose u = x and dv = e^x \, dx. Then, du = - x \, dx and v = e^x. Applying the formula: e^x x \, dx = e^x x - e^x (- x) \, dx e^x x \, dx = e^x x + e^x x \, dx Step 3: Substitute the result from Step 2 back into equation (*) from Step 1. I = e^x x - (e^x x + e^x x \, dx) I = e^x x - e^x x - e^x x \, dx Step 4: Notice that the original integral I appears on the right side. Move it to the left side to solve for I. I + e^x x \, dx = e^x x - e^x x 2I = e^x x - e^x x Step 5: Solve for I and add the constant of integration. I = (e^x x - e^x x)/(2) + C I = (e^x ( x - x))/(2) + C The integral is (e^x ( x - x))/(2) + C. Send me the next one 📸