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
To evaluate the integral y e^-x\,dx, we treat y as a constant because the integration is with respect to x. Step 1: Factor out the constant y. y e^-x\,dx = y e^-x\,dx Step 2: Integrate e^-x with respect to x. We can use a substitution for this part. Let u = -x. Then, differentiate u with respect to x: (du)/(dx) = -1 So, du = -dx, which means dx = -du. Substitute u and dx into the integral: e^-x\,dx = e^u (-du) = - e^u\,du The integral of e^u is e^u. e^u\,du = -e^u + C_1 Now, substitute back u = -x: -e^-x + C_1 Step 3: Combine the results. Multiply the result from Step 2 by y: y(-e^-x + C_1) = -y e^-x + yC_1 Since y is a constant, yC_1 is also an arbitrary constant, which we can denote as C. -y e^-x + C The final answer is -y e^-x + C. 3 done, 2 left today. You're making progress.