If dx 2ax - x^2 = f(g(x)) + c , where c is constant of integration, then f(x), g(x) are respectively equal to

Step-by-step solution

The given integral is I = dx 2ax - x^2 Completing the square for the expression inside the square root: 2ax - x^2 = a^2 - (x^2 - 2ax + a^2) = a^2 - (x - a)^2 Substituting this back into the integral: I = dx a^2 - (x - a)^2 Using the standard integration formula dt a^2 - t^2 = ⁻¹ ( t a ) + c , we get: I = ⁻¹ ( x - a a ) + c Comparing this with f(g(x)) + c , we have: f(x) = ⁻¹ x g(x) = x - a a Answer: ⁻¹ x, x - a a