JEE Advanced 2022 Paper 2Question 1
The greatest integer less than or equal to $$\int_{1}^{2}\log_{2}(x^{3}+1)\,dx+\int_{1}^{\log_{2}9}(2^{x}-1)^{\frac{1}{3}}\,dx$$ is ___________.
Definite Integration
Answer$5$
Solution
Concept. $y=(2^x-1)^{1/3}$ and $x=\log_2(y^3+1)$ are inverse functions. So the first integral is the area to the left of the curve and the second is the area below it; together they fill a rectangle minus a square. Formulas used. (1) If $g=f^{-1}$, then $\int_a^bf(x)\,dx+\int_{f(a)}^{f(b)}g(y)\,dy=bf(b)-af(a)$ ($f$ increasing)Step 1 - Identify. $f(x)=(2^x-1)^{1/3}$, $f(1)=1$, $f(\log_29)=2$, and $g(y)=\log_2(y^3+1)$.$I_1=\int_1^2g(y)\,dy$ (left of the curve), $I_2=\int_1^{\log_29}f(x)\,dx$ (below the curve). Step 2 - Use (1). $$I_1+I_2=2\log_29-1\cdot1=4\log_23-1.$$ Step 3 - Estimate. $\log_23\approx1.585$, so the sum $\approx5.34$. Greatest integer $=5$. Answer: 5