If ₀^1 4x¹¹ + 35x^6 (x^7 + 2x^5 + 5)^2 dx = 1 p , then the value of p is

Step-by-step solution

Let I = ₀^1 4x¹¹ + 35x^6 (x^7 + 2x^5 + 5)^2 dx Dividing the numerator and the denominator by x¹⁴ (which means factoring out x^7 from inside the squared denominator), we get: I = ₀^1 4x⁻³ + 35x⁻⁸ (1 + 2x⁻² + 5x⁻⁷)^2 dx Let t = 1 + 2x⁻² + 5x⁻⁷ Differentiating with respect to x : dt = (-4x⁻³ - 35x⁻⁸) dx -dt = (4x⁻³ + 35x⁻⁸) dx As x 0^+ , t As x 1 , t 1 + 2(1) + 5(1) = 8 Substituting these into the integral: I = _ ^8 -dt t^2 = ₈^ t⁻² dt I = [ - 1 t ]₈^ = 0 - (- 1 8 ) = 1 8 Given I = 1 p , we find p = 8 . Answer: 8