Integral Calculus
King's rule for symmetric integral; root identification
MMTS_Full_Test_16
Grade 12
Question:
The equation $1012x^{2023}-12138x^{2022}-119x+714=0$ has a root in $(a^{1/2022},b^{1/3})$; $a,b\in\mathbb{N}\geq2$. The value of $4\displaystyle\int_{\sqrt{a}}^{b^{1/3}}\frac{x\cos x^2}{\cos x^2+\cos(263-x^2)}\,dx$ is
(A) 25
(B) 35
(C) 40
(D) 45
Step-by-Step Solution
Key Concept: Factor polynomial: sign change between $x=6$ and $x=12$ → $a=119$ ($\sqrt{a}=\sqrt{119}$), $b=1728$ ($b^{1/3}=12$). Integral via King's rule: $I+J=\int_{\sqrt{119}}^{12}x\,dx=25/2$, $J=I\Rightarrow 4I=25$.
$4I=25$.
Correct Answer: (A) 25