MHB How Do You Solve for C in This Integral Equation?

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The integral equation provided is $\int_1^{f(x)}g(t) \,dt =\frac{1}{3}\left(x^{3/2}-8\right)$, with the relationship $f^{-1}(x)=g(x)$. By applying the fundamental theorem of calculus, the derivative leads to the equation $g(f(x))\cdot f'(x)=\frac{1}{2}\sqrt{x}$, which simplifies to $f'(x)=\frac{1}{2\sqrt{x}}$. Integrating this gives $f(x)=\sqrt{x}+C$. The discussion seeks a method to determine the constant $C$.
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Have added attachment. Can anyone show me how to approach this problem? Thank you...
 

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Hi scottshannon!

We have $\int_1^{f(x)}g(t) \,dt =\frac{1}{3}\left(x^{3/2}-8\right)$ with $f^{-1}(x)=g(x)$. Applying the fundamental theorem of calculus on both sides:
$$g(f(x))\cdot f'(x)=\frac{1}{2}\sqrt{x}$$
$$x \cdot f'(x) =\frac{1}{2}\sqrt{x}$$
$$f'(x)=\frac{1}{2\sqrt{x}}$$
Solving the resulting by integrating:
$$f(x)=\sqrt{x}+C$$

Now, how may we solve for $C$?
 
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