Integral of an inverse Let $f(x)=x^3−2x^2+5$. Then find the integral
$$\int_{37}^{149} \! f^{-1}(x) \, \mathrm{d}x$$ I know the inverse theorem for differentiation.( I don't think we can apply it here). Is there other theorem for integration.(I am not finding the inverse and then integrating).

Here $f(x)$ is a cubic polynomial. $f'(x)=x(3x-4)>0$ for $x>37$. So for $x>37, f$ is increasing and hence it is a one to one and onto function from $37$ to $149$. So $f^{-1}$ exists on $[37,149]$. Now use change of variables.
Let $x=f(t)$ for $x\in [37,149]$. Then $f^{-1}(x)=t$. Then, $dx=f'(t)dt$. So, $f^{-1}(x) dx=tf'(t)dt$. 
Hence, $\displaystyle \int_{37}^{149} f^{-1}(x) dx=\int_{f^{-1}(37)}^{f^{-1}(149)} tf'(t)dt$.
