if $ f(x)=x+\cos x $ then find $ \int_0^\pi (f^{-1}(x))\text{dx} $? I would be interest to show :
if $ f(x)=x+\cos x $  then find $ \int_0^\pi (f^{-1}(x))\text{dx} $ ?
my  second question that's make me  a problem is that :
what is :$ f^{-1}(\pi) $ ?
I would be interest for any replies or any comments .

$$\displaystyle \int_{a}^{b}f(x)dx+\int_{f(a)}^{f(b)}f^{-1}(x)dx=bf(b)-af(a)$$
$$\displaystyle \int_{a}^{b}(x+\cos x)dx+\int_{0}^{\pi}f^{-1}(x)dx=b\pi$$
$$ f(b)=\pi\Rightarrow b+\cos b=\pi\Rightarrow b\approx 3.88$$
$$f(a)=0\Rightarrow a+\cos a=0\Rightarrow a\approx -0.74$$
$$\displaystyle \int_{0}^{\pi}f^{-1}(x)dx=b\pi-\frac{b^{2}-a^{2}}{2}+(\sin( b)-\sin(a))$$
