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: Rewrite the integrand using exponent rules. The integral is given as _x=1^x=2 ((1)/(x) - x^2) dx. We can rewrite (1)/(x) as x^-1. _1^2 (x^-1 - x^2) dx Step 2: Integrate each term. The integral of x^-1 is |x|. The integral of x^n is x^n+1n+1. So, the integral of -x^2 is -x^2+12+1 = -(x^3)/(3). [|x| - (x^3)/(3)]_1^2 Step 3: Evaluate the definite integral using the limits of integration. Substitute the upper limit (x=2) and subtract the result of substituting the lower limit (x=1). (|2| - (2^3)/(3)) - (|1| - (1^3)/(3)) ((2) - (8)/(3)) - ((1) - (1)/(3)) Since (1) = 0: ((2) - (8)/(3)) - (0 - (1)/(3)) (2) - (8)/(3) + (1)/(3) (2) - (7)/(3) The final answer is (2) - (7)/(3). Send me the next one 📸