Question

Assertion (A): \[ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{(\sin x)^{\sqrt{2}}}{(\sin x)^{\sqrt{2}} + (\cos x)^{\sqrt{2}}} \, dx = \frac{\pi}{12} \]

Reason (R): \[ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{f(x)}{f(x) + f\left(\frac{\pi}{2}-x\right)} \, dx = \frac{\pi}{12} \]

• A. A is true, R is true and R is the correct explanation of A
• B. A is true, R is true but R is not the correct explanation of A
• C. A is true, R is false
• D. A is false, R is true

Hint:

When solving integrals with symmetry, consider using substitution or symmetry properties of trigonometric functions. In this case, the substitution \( x \to \frac{\pi}{2} - x \) simplifies the integral and leads to the desired result.

Answer 1

The Correct Option is A

We are given the assertion \( A \) and the reason \( R \). We need to verify whether the given integrals are true and whether the reason provided is the correct explanation of the assertion. Assertion (A): \[ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{(\sin x)^{\sqrt{2}}}{(\sin x)^{\sqrt{2}} + (\cos x)^{\sqrt{2}}} \, dx = \frac{\pi}{12} \] This integral represents a well-known identity in trigonometry, where such integrals evaluate to \( \frac{\pi}{12} \) for specific powers of \( \sqrt{2} \). Hence, the assertion (A) is true. Reason (R): The second integral in the reason is: \[ I_2 = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{f(x)}{f(x) + f\left( \frac{\pi}{2}-x \right)} \, dx \] By using symmetry and the fact that \( f(x) = (\sin x)^{\sqrt{2}} \), this integral simplifies to the same form as the one in assertion (A), and it evaluates to \( \frac{\pi}{12} \). Thus, the reason (R) is a correct explanation for assertion (A). Since both the assertion and the reason are true, and the reason correctly explains the assertion, the correct answer is option (1).