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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To solve the integral, we will use the method of partial fraction decomposition. Step 1: Factor the denominator. The denominator is x^3 + x^2 - 2x. x^3 + x^2 - 2x = x(x^2 + x - 2) Factor the quadratic term x^2 + x - 2: x^2 + x - 2 = (x+2)(x-1) So, the fully factored denominator is: x(x+2)(x-1) Step 2: Set up the partial fraction decomposition. We express the integrand as a sum of simpler fractions: (2x^2 + 5x - 1)/(x(x+2)(x-1)) = (A)/(x) + (B)/(x+2) + (C)/(x-1) Step 3: Solve for the constants A, B, and C. Multiply both sides by x(x+2)(x-1): 2x^2 + 5x - 1 = A(x+2)(x-1) + Bx(x-1) + Cx(x+2) To find A, set x=0: 2(0)^2 + 5(0) - 1 = A(0+2)(0-1) -1 = A(2)(-1) -1 = -2A A = (1)/(2) To find B, set x=-2: 2(-2)^2 + 5(-2) - 1 = B(-2)(-2-1) 2(4) - 10 - 1 = B(-2)(-3) 8 - 10 - 1 = 6B -3 = 6B B = -(3)/(6) = -(1)/(2) To find C, set x=1: 2(1)^2 + 5(1) - 1 = C(1)(1+2) 2 + 5 - 1 = C(1)(3) 6 = 3C C = 2 Step 4: Rewrite the integral using the partial fractions. (2x^2 + 5x - 1)/(x^3 + x^2 - 2x) \, dx = ( (1/2)/(x) - (1/2)/(x+2) + (2)/(x-1) ) \, dx Step 5: Integrate each term. = (1)/(2) (1)/(x) \, dx - (1)/(2) (1)/(x+2) \, dx + 2 (1)/(x-1) \, dx = (1)/(2) |x| - (1)/(2) |x+2| + 2 |x-1| + C Step 6: Combine the logarithmic terms using logarithm properties ( a - b = (a)/(b) and c a = a^c). = (1)/(2) ( |x| - |x+2| ) + |(x-1)^2| + C = (1)/(2) | (x)/(x+2) | + |(x-1)^2| + C The final answer is (1)/(2) | (x)/(x+2) | + |(x-1)^2| + C. Send me the next one 📸