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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You're asking about how to perform the integration of the expression (-12t + 8) with respect to t. Step 1: Apply the power rule for integration. The power rule states that x^n \, dx = x^n+1n+1 + C. For a constant k, k \, dx = kx + C. We integrate each term separately: (-12t + 8) \, dt = -12t \, dt + 8 \, dt Step 2: Integrate each term. For the first term, -12t \, dt: Here, t has a power of 1. So, n=1. -12t^1 \, dt = -12 t^1+11+1 = -12 (t^2)/(2) = -6t^2 For the second term, 8 \, dt: This is a constant. 8 \, dt = 8t Step 3: Combine the integrated terms. So, the indefinite integral is -6t^2 + 8t + C. Since we are evaluating a definite integral from 0 to 1, we don't need the constant C. We write the result in the evaluation bracket: [ -6t^2 + 8t ]_0^1 Step 4: Apply the Fundamental Theorem of Calculus. This means we evaluate the antiderivative at the upper limit (t=1) and subtract its value at the lower limit (t=0). (-6(1)^2 + 8(1)) - (-6(0)^2 + 8(0)) (-6 + 8) - (0 + 0) 2 - 0 2 The value of the integral is 2. Last free one today — make it count tomorrow, or type /upgrade for unlimited.