JEE Main 2024 — Definite Integration Question with Solution

Equation of CE

$$\begin{array}{rl}& y-1=-(x-3)\\ & x+y=4\end{array}$$

orthocentre lies on the line $$x+y=4$$

so, $$a+b=4$$

I_1=\int_\limits a^b x \sin (x(4-x)) d x\quad ..... (i)

Using king rule

I_1=\int_\limits a^b(4-x) \sin (x(4-x)) d x\quad .... (ii)

\begin{aligned} & \text { (i) }+ \text { (ii) } \\ & 2 \mathrm{I}_1=\int_\limits{\mathrm{a}}^{\mathrm{b}} 4 \sin (\mathrm{x}(4-\mathrm{x})) \mathrm{dx} \\ & 2 \mathrm{I}_1=4 \mathrm{I}_2 \\ & \mathrm{I}_1=2 \mathrm{I}_2 \\ & \frac{\mathrm{I}_1}{\mathrm{I}_2}=2 \\ & \frac{36 \mathrm{I}_1}{\mathrm{I}_2}=72 \end{aligned}