If f(x) = x (x^2 + 4)(x^2 + 9) , dx and f(0) = 1 5 ( 2 3 ) , then f(1) =

Step-by-step solution

Let x^2 = t , then 2x , dx = dt , which gives x , dx = dt 2 . Substituting this into the integral, we get: f(x) = 1 2 dt (t + 4)(t + 9) Using partial fractions: f(x) = 1 2 1 5 ( 1 t + 4 - 1 t + 9 ) dt f(x) = 1 10 ( |t + 4| - |t + 9| ) + C f(x) = 1 10 ( t + 4 t + 9 ) + C Substituting t = x^2 : f(x) = 1 10 ( x^2 + 4 x^2 + 9 ) + C Given f(0) = 1 5 ( 2 3 ) , we substitute x = 0 : f(0) = 1 10 ( 4 9 ) + C f(0) = 1 10 ( ( 2 3 )^2 ) + C f(0) = 1 5 ( 2 3 ) + C Comparing with the given value, we get C = 0 . Thus, the functio