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
Here's the solution for question 2: Step 1: Rewrite the integrand by dividing each term in the numerator by the denominator. _2^4 ((x^3+3)/(x^2))dx = _2^4 ((x^3)/(x^2) + (3)/(x^2))dx = _2^4 (x + 3x^-2)dx Step 2: Integrate each term with respect to x. (x + 3x^-2)dx = x^1+11+1 + 3x^-2+1-2+1 + C = (x^2)/(2) + 3x^-1-1 + C = (x^2)/(2) - (3)/(x) + C Step 3: Evaluate the definite integral using the limits of integration from 2 to 4. [ (x^2)/(2) - (3)/(x) ]_2^4 = ((4^2)/(2) - (3)/(4)) - ((2^2)/(2) - (3)/(2)) Step 4: Simplify the expression. = ((16)/(2) - (3)/(4)) - ((4)/(2) - (3)/(2)) = (8 - (3)/(4)) - (2 - (3)/(2)) = ((32)/(4) - (3)/(4)) - ((4)/(2) - (3)/(2)) = (29)/(4) - (1)/(2) = (29)/(4) - (2)/(4) = (27)/(4) The value of the integral is (27)/(4). That's 2 down. 3 left today — send the next one.