Let f(x) be a function defined for x > 0 such that f(x) = 3x + 8 (x^4 + x + 2)^ 5/4 dx . If f(1) = 2 2 + 3 , then the value of _ x f(x) is equal to ________ .

Step-by-step solution

Given f(x) = 3x + 8 (x^4 + x + 2)^ 5/4 dx Extract x^4 from the expression inside the fractional power: (x^4 + x + 2)^ 5/4 = (x^4(1 + x⁻³ + 2x⁻⁴) )^ 5/4 = x^5 (1 + x⁻³ + 2x⁻⁴)^ 5/4 The integral becomes: f(x) = 3x + 8 x^5 (1 + x⁻³ + 2x⁻⁴)^ 5/4 dx f(x) = (3x⁻⁴ + 8x⁻⁵) (1 + x⁻³ + 2x⁻⁴)^ -5/4 dx Let t = 1 + x⁻³ + 2x⁻⁴ . Differentiating both sides with respect to x : dt = (-3x⁻⁴ - 8x⁻⁵) dx = -(3x⁻⁴ + 8x⁻⁵) dx (3x⁻⁴ + 8x⁻⁵) dx = -dt Substituting this into the integral: f(x) = -t^ -5/4 dt = - t^ -1/4 -1/4 + C = 4t^ -1/4 +