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.
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Here are the solutions for the integration problems: 18. To evaluate the integral (2x - x^2)\,dx: Step 1: Apply the sum/difference rule for integration. (2x - x^2)\,dx = 2x\,dx - x^2\,dx Step 2: Apply the power rule for integration x^n\,dx = x^n+1n+1 + C. 2x\,dx = 2 · x^1+11+1 + C_1 = 2 · (x^2)/(2) + C_1 = x^2 + C_1 x^2\,dx = x^2+12+1 + C_2 = (x^3)/(3) + C_2 Step 3: Combine the results and add the constant of integration C. (2x - x^2)\,dx = x^2 - (x^3)/(3) + C The integral is x^2 - (x^3)/(3) + C. 19. To evaluate the integral (t^4 - 3t^2 + 1)/(t^4)\,dt: Step 1: Divide each term in the numerator by the denominator to simplify the expression. (t^4 - 3t^2 + 1)/(t^4) = (t^4)/(t^4) - (3t^2)/(t^4) + (1)/(t^4) = 1 - 3t^-2 + t^-4 Step 2: Integrate the simplified expression term by term. (1 - 3t^-2 + t^-4)\,dt = 1\,dt - 3t^-2\,dt + t^-4\,dt Step 3: Apply the power rule for integration t^n\,dt = t^n+1n+1 + C. 1\,dt = t + C_1 3t^-2\,dt = 3 · t^-2+1-2+1 + C_2 = 3 · t^-1-1 + C_2 = -3t^-1 + C_2 t^-4\,dt = t^-4+1-4+1 + C_3 = t^-3-3 + C_3 = -(1)/(3)t^-3 + C_3 Step 4: Combine the results and add the constant of integration C. (t^4 - 3t^2 + 1)/(t^4)\,dt = t - (-3t^-1) + (-(1)/(3)t^-3) + C = t + 3t^-1 - (1)/(3)t^-3 + C Step 5: Express with positive exponents. = t + (3)/(t) - (1)/(3t^3) + C The integral is t + (3)/(t) - (1)/(3t^3) + C. That's 2 down. 3 left today — send the next one.