JEE Advanced 2022 · Paper 2, Mathematics question 3

Inverse functions. Let $f(x)=\log_2(x^3+1)$ on $[1,2]$; it is increasing, with $f(1)=1$ and $f(2)=\log_29$. Solving $y=\log_2(x^3+1)$ for $x$ gives
$$f^{-1}(y)=(2^y-1)^{1/3}.$$
So the second integral is $\int_{f(1)}^{f(2)}f^{-1}(y)\,dy$.

Area identity. For an increasing function,
$$\int_a^bf(x)\,dx+\int_{f(a)}^{f(b)}f^{-1}(y)\,dy=bf(b)-af(a).$$

Value.
$$2\log_29-1=2(3.17)-1\approx5.34.$$
The greatest integer is 5.