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).

For monotonically increasing function,
$$\int_{f(a)}^{f(b)}f^{-1}(x)dx=bf(b)-af(a)-\int_a^bf(x)dx$$
Here,a=4,b=6,
By the way,this result can be proved by observing the graph of a monotonic function.The function you gave is monotonically increasing in given domain.
