Evaluate the definite integral.

Step 1: Evaluate the definite integral. The integral to evaluate is _1^p (px-1) dx. First, find the antiderivative of px-1 with respect to x: (px-1) dx = p(x^2)/(2) - x + C Now, apply the limits of integration from 1 to p: [ (px^2)/(2) - x ]_1^p = ( (p(p)^2)/(2) - p ) - ( (p(1)^2)/(2) - 1 ) = ( (p^3)/(2) - p ) - ( (p)/(2) - 1 ) = (p^3)/(2) - p - (p)/(2) + 1 Combine the terms involving p: = (p^3)/(2) - (2p)/(2) - (p)/(2) + 1 = (p^3)/(2) - (3p)/(2) + 1 Step 2: Set the evaluated integral equal to 0. We are given that _1^p (px-1) dx = 0. So, we set the expression from Step 1 to 0: (p^3)/(2) - (3p)/(2) + 1 = 0 Multiply the entire equation by 2 to eliminate the denominators: p^3 - 3p + 2 = 0 Step 3: Solve the cubic equation for p. We look for integer roots by testing divisors of the constant term (2), which are ± 1, ± 2. Test p=1: (1)^3 - 3(1) + 2 = 1 - 3 + 2 = 0 Since p=1 satisfies the equation, (p-1) is a factor of the polynomial. We can use polynomial division or synthetic division to find the other factors. Using synthetic division with root 1: ` 1 | 1 0 -3 2 | 1 1 -2 ---------------- 1 1 -2 0 ` The quotient is p^2 + p - 2. So, the cubic equation can be factored as: (p-1)(p^2 + p - 2) = 0 Now, factor the quadratic term p^2 + p - 2. We need two numbers that multiply to -2 and add to 1. These numbers are 2 and -1. p^2 + p - 2 = (p+2)(p-1) Substitute this back into the factored cubic equation: (p-1)(p+2)(p-1) = 0 (p-1)^2(p+2) = 0 Step 4: Determine the values of p. From (p-1)^2 = 0, we get p-1 = 0, which means p=1. From (p+2) = 0, we get p=-2. Thus, the values of p are 1 and -2. The final answer is p=1, p=-2. Send me the next one 📸