### Related Formula int_a^b f(x) dx = int_a^b f(a+b-x) dx quad text(King's Rule)$\int_a^b f(x) dx = \int_a^b f(a+b-x) dx \quad \text{(King's Rule)}$ ### Core Logic First, find the orthocentre (a,b)$(a,b)$ of Delta ABC$\Delta ABC$ with vertices A(1, 2)$A(1, 2)$, B(2, 3)$B(2, 3)$, and C(3, 1)$C(3, 1)$. Slope of AB = frac3-22-1 = 1$AB = \frac{3-2}{2-1} = 1$. The altitude from C$C$ onto AB$AB$ must be perpendicular to AB$AB$, so its slope is -1$-1$. Equation of altitude from C(3,1)$C(3,1)$: y - 1 = -1(x - 3) Rightarrow x + y = 4$y - 1 = -1(x - 3) \Rightarrow x + y = 4$ The orthocentre (a, b)$(a, b)$ lies on all altitudes, including this one. Thus, it satisfies a + b = 4$a + b = 4$. ### Step 1: Applying Definite Integral Properties Given I_1 = int_a^b x sin(4x-x^2) dx$I_1 = \int_a^b x \sin(4x-x^2) dx$, let's rewrite the argument of sine: 4x - x^2 = x(4-x)$4x - x^2 = x(4-x)$ Apply King's Rule replacing x$x$ with (a+b-x)$(a+b-x)$. Since we proved a+b = 4$a+b = 4$, substitute x$x$ with (4-x)$(4-x)$: I_1 = int_a^b (4-x) sin((4-x)(4 - (4-x))) dx$I_1 = \int_a^b (4-x) \sin((4-x)(4 - (4-x))) dx$ I_1 = int_a^b (4-x) sin((4-x)x) dx$I_1 = \int_a^b (4-x) \sin((4-x)x) dx$ I_1 = int_a^b (4-x) sin(4x-x^2) dx$I_1 = \int_a^b (4-x) \sin(4x-x^2) dx$ ### Step 2: Evaluating the Integral Ratio Expand the newly formed integral: I_1 = 4 int_a^b sin(4x-x^2) dx - int_a^b x sin(4x-x^2) dx$I_1 = 4 \int_a^b \sin(4x-x^2) dx - \int_a^b x \sin(4x-x^2) dx$ Notice that the second term is I_1$I_1$ and the first integral is I_2$I_2$: I_1 = 4I_2 - I_1 Rightarrow 2I_1 = 4I_2 Rightarrow fracI_1I_2 = 2$I_1 = 4I_2 - I_1 \Rightarrow 2I_1 = 4I_2 \Rightarrow \frac{I_1}{I_2} = 2$ ### Step 3: Final Output Evaluation We need the value of 36 fracI_1I_2$36 \frac{I_1}{I_2}$: 36 times 2 = 72$36 \times 2 = 72$ ### Pattern Recognition Whenever you see int_a^b x cdot f(x(a+b-x)) dx$\int_a^b x \cdot f(x(a+b-x)) dx$, immediately apply King's Rule to factor out x$x$. You rarely need the individual values of the integration bounds, only their sum. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Maths: Straight Lines Class 12 Maths: Definite Integration