Step 1: Approach
Check each option by differentiating, using a simple test function.
Step 2: Test function
Let $f(x)=x$, so $g(x)=\dfrac{x^2}{2}$ and $f^{-1}(x)=x$. The true integral is $\int x\,dx=\dfrac{x^2}{2}$.
Step 3: Evaluate option (C)
$x\cdot x-g(x)=x^2-\dfrac{x^2}{2}=\dfrac{x^2}{2}$. This matches.
Step 4: Rule out others
Option (A) gives $x^2$, which is wrong. Option (D) gives $g^{-1}(x)=\sqrt{2x}$, which is wrong. So (C) is the correct formula.
Final Answer:
The integral of the inverse function is $xf^{-1}(x)-g(f^{-1}(x))+c$, option (C).
\[ \boxed{xf^{-1}(x)-g(f^{-1}(x))+c} \]