This question looks a bit like inverse chain rule but inverse chain rule require the kernel of the integral to contain a product of two functions, one is a composite fg(x) and the other is the differential of the inner function g’(x). If you see this form then you can use inverse chain rule directly. Often, g’(x) can be generated algebraically but this is usually only helpful if the g’(x) is a constant. Assuming g’(x) is a constant k, Then you can just say k . 1/k = 1 and put k outside the integral and 1/k inside but the g’(x) is itself a function of x then you would need to put something involving x outside the integral (which is absolutely not ok).
Hence, in this case you need to expand the bracket and integrate term by term.