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
Welcome back — missed you this week. To integrate _1^e x x \, dx, we use integration by parts, which states u \, dv = uv - v \, du. Step 1: Choose u and dv. Let u = x and dv = x \, dx. Step 2: Find du and v. Differentiate u to find du: du = (1)/(x) \, dx Integrate dv to find v: v = x \, dx = (x^2)/(2) Step 3: Apply the integration by parts formula. x x \, dx = ( x) ((x^2)/(2)) - ((x^2)/(2)) ((1)/(x)) \, dx x x \, dx = (x^2)/(2) x - (x)/(2) \, dx Step 4: Evaluate the remaining integral. (x)/(2) \, dx = (1)/(2) x \, dx = (1)/(2) ((x^2)/(2)) = (x^2)/(4) Step 5: Combine the results for the indefinite integral. x x \, dx = (x^2)/(2) x - (x^2)/(4) Step 6: Evaluate the definite integral from 1 to e. _1^e x x \, dx = [ (x^2)/(2) x - (x^2)/(4) ]_1^e Substitute the upper limit x=e: ( (e^2)/(2) e - (e^2)/(4) ) = ( (e^2)/(2)(1) - (e^2)/(4) ) = (e^2)/(2) - (e^2)/(4) = (2e^2 - e^2)/(4) = (e^2)/(4) Substitute the lower limit x=1: ( (1^2)/(2) 1 - (1^2)/(4) ) = ( (1)/(2)(0) - (1)/(4) ) = 0 - (1)/(4) = -(1)/(4) Subtract the lower limit result from the upper limit result: (e^2)/(4) - (-(1)/(4)) = (e^2)/(4) + (1)/(4) = (e^2+1)/(4) The final answer is (e^2+1)/(4). Send me the next one 📸