1 x a x-x^2 d x is

Step-by-step solution

Let I = 1 x ax - x^2 dx . Rewrite the integral as I = 1 x x(a - x) dx . Divide the numerator and denominator inside the square root by x^2 : I = 1 x x a - x x dx = 1 x^2 a x - 1 dx . Let t = a x - 1 . Then t^2 = a x - 1 , which implies t^2 + 1 = a x , so x = a t^2 + 1 . Differentiating x with respect to t : dx = a (-1) (t^2 + 1)⁻² (2t) dt = -2at (t^2 + 1)^2 dt . Substitute x and dx into the integral: I = 1 ( a t^2 + 1 )^2 t ( -2at (t^2 + 1)^2 ) dt . I = (t^2 + 1)^2 a^2 t -2at (t^2 + 1)^2 dt = -2 a dt . I = - 2 a t