If dx x^ 7/2 (x^4 + 1)^ 3/8 = m ( x^4 + 1 x^4 )^n + c , where c is a constant of integration, then the value of n m is...

Step-by-step solution

Given integral is I = dx x^ 7/2 (x^4 + 1)^ 3/8 Factoring out x^4 from the term (x^4 + 1)^ 3/8 , we get: I = dx x^ 7/2 (x^4)^ 3/8 (1 + 1 x^4 )^ 3/8 I = dx x^ 7/2 x^ 3/2 (1 + x⁻⁴)^ 3/8 I = dx x^5 (1 + x⁻⁴)^ 3/8 Let 1 + x⁻⁴ = t Differentiating both sides, we get: -4x⁻⁵ dx = dt dx x^5 = - dt 4 Substituting this into the integral: I = - 1 4 t^ -3/8 dt I = - 1 4 ( t^ 5/8 5/8 ) + c I = - 2 5 t^ 5/8 + c Substituting back t = 1 + x⁻⁴ = x^4 + 1 x^4 : I = - 2 5 ( x^4 + 1 x^4 )^ 5/8 + c Comparing this with the given expression