So I have a function $$ f(x)=x^{3}-12x^{2}+69x+6 $$
The question askes me to find the relationship between $$\int_{1}^{2} f(x)dx$$ $$and$$ $$\int_{64}^{104} f^{-1}(x)dx$$
And compute the value of $\int_{64}^{104} f^{-1}(x)dx$
I found out that $f(1) = 64$ and $f(2) = 104 $
But not sure how do I proceed.
I could not find how this is related to the course material presented this week, and am completely lost to be honest. Any help would be appreciated
Hint.
By Laisant's formula, if $f(a)=c$ and $f(b)=d$, then $$ \int_{c}^df^{-1}(x)dx+\int_a^bf(x)dx=bd-ac $$
Notes.
When $f$ is differentiable, one can prove the result by change of variables: $$ \int_c^d f^{-1}(y)dy=\int_a^b f^{-1}(f(x))f'(x)dx=\int_a^b xf'(x)dx=xf(x)|_{a}^b-\int_a^bf(x)dx $$
A proof without words: