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    #1
    Have added attachment. Can anyone show me how to approach this problem? Thank you....

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    #2
    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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