>I also know that the question to check for invertibility means to check if the function is 1-to-1 and onto.

That is not what the question is asking. They are asking you whether d/dx and ∫ are inverse **to each other**, not just whether they are invertible at all. To say that two functions f,g are inverse to each other means that f∘g and g∘f are both the identity map on their respective domains. The answer is showing that ∫(d/dx(1))≠1, ie, ∫∘d/dx is not the identity map.