If 1 ^3 x 2x , dx = p( ^2 x + q) x + c , then the values of p and q respectively are

Step-by-step solution

Let I = 1 ^3 x 2x dx Using 2x = 2 x x , we get: I = 1 ^3 x 2 x x dx I = 1 2 1 ^ 7/2 x ^ 1/2 x dx Dividing the numerator and the denominator by ^4 x , we get: I = 1 2 ^4 x ^ 1/2 x dx Let x = t ^2 x dx = dt I = 1 2 1 + t^2 t dt I = 1 2 (t^ -1/2 + t^ 3/2 ) dt I = 1 2 ( t^ 1/2 1/2 + t^ 5/2 5/2 ) + c I = 1 2 ( 2 t + 2 5 t^ 5/2 ) + c I = 2 t ( 1 + 1 5 t^2 ) + c I = 2 5 (t^2 + 5) t + c Substituting t = x : I = 2 5 ( ^2 x + 5) x + c Comparing this with p( ^2 x + q) x + c , we get: p = 2 5 and q = 5 Answer: p = 2 5 , q = 5