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
Let's integrate this expression. Step 1: Expand the integrand. We use the formula (a+b)^2 = a^2 + 2ab + b^2. (e^x + 2e^-x)^2 = (e^x)^2 + 2(e^x)(2e^-x) + (2e^-x)^2 = e^2x + 4e^x-x + 4e^-2x = e^2x + 4e^0 + 4e^-2x = e^2x + 4 + 4e^-2x Step 2: Integrate each term. We integrate term by term: (e^2x + 4 + 4e^-2x)\,dx = e^2x\,dx + 4\,dx + 4e^-2x\,dx = (1)/(2)e^2x + 4x + 4(-(1)/(2))e^-2x = (1)/(2)e^2x + 4x - 2e^-2x Step 3: Apply the limits of integration. Now we evaluate the definite integral from 0 to 2: [ (1)/(2)e^2x + 4x - 2e^-2x ]_0^ 2 Step 4: Evaluate at the upper limit (x = 2). (1)/(2)e^2 2 + 4 2 - 2e^-2 2 Using the logarithm property a b = (b^a) and e^ k = k: = (1)/(2)e^(2^2) + 4 2 - 2e^(2^-2) = (1)/(2)e^ 4 + 4 2 - 2e^(1/4) = (1)/(2)(4) + 4 2 - 2((1)/(4)) = 2 + 4 2 - (1)/(2) = (4)/(2) - (1)/(2) + 4 2 = (3)/(2) + 4 2 Step 5: Evaluate at the lower limit (x = 0). (1)/(2)e^2(0) + 4(0) - 2e^-2(0) = (1)/(2)e^0 + 0 - 2e^0 = (1)/(2)(1) - 2(1) = (1)/(2) - 2 = -(3)/(2) Step 6: Subtract the lower limit value from the upper limit value. ( (3)/(2) + 4 2 ) - ( -(3)/(2) ) = (3)/(2) + 4 2 + (3)/(2) = (6)/(2) + 4 2 = 3 + 4 2 The final answer is 3 + 4 2. That's 2 down. 3 left today — send the next one.