Integral Calculus
Integral Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
(A) If $f(x)$ is an integrable function for $x \in \left[\frac{\pi}{6}, \frac{\pi}{3}\right]$ and $I_1 = \int_{\pi/6}^{\pi/3} \sec^2 \theta(2\sin 20)d\theta$ and $I_2 = \int_{\pi/6}^{\pi/3} \cos e^2 \theta(2\sin 20)d\theta$, then $I_1/I_2$
Step-by-Step Solution
Key Concept: Decompose a rational function by strategically splitting the numerator to match partial denominators and apply substitution techniques.
The integral $I = \int \frac{(x^2 + x + 1)dx}{(x^2 + x + 1)(x^2 - x + 1)}$ is decomposed by splitting the numerator into $(x^2 - x + 1) + (x^2 + x + 1) - x$ in the standard form. This yields three integrals: $I_1$ (with numerator $xdx$), $I_2$ (with numerator $dx$), and $I_3$ (with numerator $xdx$). For $I_3$, substituting $x^2 = t$ transforms it to a standard arctangent form: $I_3 = \frac{1}{\sqrt{3}}\tan^{-1}\left(\frac{2x^2 + 1}{\sqrt{3}}\right)$.
Correct Answer: [A-q] [B-r, s] [C-p] [D-p]