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 solve the definite integral _0^1 t (t)\,dt, we will use integration by parts. Step 1: Recall the integration by parts formula. The formula for integration by parts is u\,dv = uv - v\,du. Step 2: Choose u and dv. Let u = t and dv = (t)\,dt. Step 3: Find du and v. Differentiate u to find du: du = dt Integrate dv to find v: v = (t)\,dt = (t) Step 4: Apply the integration by parts formula to the indefinite integral. t (t)\,dt = t (t) - (t)\,dt Step 5: Solve the remaining integral. (t)\,dt = (t) Step 6: Substitute the result back into the expression from Step 4. The indefinite integral is: t (t)\,dt = t (t) - (t) + C Step 7: Evaluate the definite integral using the limits from 0 to 1. _0^1 t (t)\,dt = [ t (t) - (t) ]_0^1 Apply the Fundamental Theorem of Calculus: = (1 · (1) - (1)) - (0 · (0) - (0)) Step 8: Calculate the values of (1), (1), (0), and (0). Recall that (x) = e^x - e^-x2 and (x) = e^x + e^-x2. (1) = e^1 - e^-12 (1) = e^1 + e^-12 (0) = e^0 - e^-02 = (1 - 1)/(2) = 0 (0) = e^0 + e^-02 = (1 + 1)/(2) = 1 Step 9: Substitute these values into the definite integral expression. = ( e - e^-12 - e + e^-12 ) - (0 · 0 - 1) = ( e - e^-1 - e - e^-12 ) - (-1) = ( -2e^-12 ) + 1 = -e^-1 + 1 = 1 - (1)/(e) The final answer is 1 - (1)/(e). Last free one today — make it count tomorrow, or type /upgrade for unlimited.