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: Simplify the integral expression. The given function is f(x) = (3x^4 - 4x^3 - x^2 + 1)/(x^2) d(2x). First, rewrite d(2x) as 2dx. f(x) = (3x^4 - 4x^3 - x^2 + 1)/(x^2) (2dx) Factor out the constant 2: f(x) = 2 (3x^4 - 4x^3 - x^2 + 1)/(x^2) dx Divide each term in the numerator by x^2: f(x) = 2 ( (3x^4)/(x^2) - (4x^3)/(x^2) - (x^2)/(x^2) + (1)/(x^2) ) dx f(x) = 2 ( 3x^2 - 4x - 1 + x^-2 ) dx Step 2: Perform the integration. Integrate each term: f(x) = 2 ( 3x^2+12+1 - 4x^1+11+1 - x^0+10+1 + x^-2+1-2+1 ) + C f(x) = 2 ( (3x^3)/(3) - (4x^2)/(2) - x - x^-11 ) + C f(x) = 2 ( x^3 - 2x^2 - x - (1)/(x) ) + C Distribute the 2: f(x) = 2x^3 - 4x^2 - 2x - (2)/(x) + C Step 3: Use the condition f(1) = 2 to find the constant C. Substitute x=1 into the expression for f(x): f(1) = 2(1)^3 - 4(1)^2 - 2(1) - (2)/(1) + C 2 = 2(1) - 4(1) - 2 - 2 + C 2 = 2 - 4 - 2 - 2 + C 2 = -6 + C Solve for C: C = 2 + 6 C = 8 Step 4: Write the complete function f(x). Substitute the value of C back into the function: f(x) = 2x^3 - 4x^2 - 2x - (2)/(x) + 8 Step 5: Calculate f(-1). Substitute x=-1 into the function: f(-1) = 2(-1)^3 - 4(-1)^2 - 2(-1) - (2)/(-1) + 8 f(-1) = 2(-1) - 4(1) - (-2) - (-2) + 8 f(-1) = -2 - 4 + 2 + 2 + 8 f(-1) = -6 + 4 + 8 f(-1) = -2 + 8 f(-1) = 6 The final answer is 6. Send me the next one 📸