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
Step 1: _1^5 2x \, dx 2x \, dx = x^2 + C [ x^2 ]_1^5 = 5^2 - 1^2 5^2 = 25, 1^2 = 1 25 - 1 = 24 _1^5 2x \, dx = 24 24 Step 2: _1^5 (2x + 3) \, dx (2x + 3) \, dx = x^2 + 3x + C [ x^2 + 3x ]_1^5 = (5^2 + 3 · 5) - (1^2 + 3 · 1) 5^2 + 3 · 5 = 25 + 15 = 40 1^2 + 3 · 1 = 1 + 3 = 4 40 - 4 = 36 _1^5 (2x + 3) \, dx = 36 36 Step 3: f(x) = x^2 - 3x + 2, _1^5 f(x) \, dx = _1^5 (x^2 - 3x + 2) \, dx (x^2 - 3x + 2) \, dx = (1)/(3) x^3 - (3)/(2) x^2 + 2x + C [ (1)/(3) x^3 - (3)/(2) x^2 + 2x ]_1^5 At x=5: (1)/(3) (125) - (3)/(2) (25) + 2(5) = (125)/(3) - (75)/(2) + 10 Common denominator 6: (125)/(3) = (250)/(6), (75)/(2) = (225)/(6), 10 = (60)/(6) (250)/(6) - (225)/(6) + (60)/(6) = (250 - 225 + 60)/(6) = (85)/(6) At x=1: (1)/(3) (1) - (3)/(2) (1) + 2(1) = (1)/(3) - (3)/(2) + 2 Common denominator 6: (1)/(3) = (2)/(6), (3)/(2) = (9)/(6), 2 = (12)/(6) (2)/(6) - (9)/(6) + (12)/(6) = (2 - 9 + 12)/(6) = (5)/(6) (85)/(6) - (5)/(6) = (80)/(6) = (40)/(3) _1^5 (x^2 - 3x + 2) \, dx = (40)/(3) (40)/(3)