AP EAMCET · Maths · Definite Integration
Given that \(\frac{d}{d x} \int_0^{\phi(x)} f(t) d t=f(\phi(x)) \phi^{\prime}(x)\). For all \(x \in\left(0, \frac{\pi}{2}\right)\), if \(\int_1^{\cos x} t^2 f(t) d t=\cos 2 x\), then \(f\left(\frac{1}{\sqrt{2}}\right)=\)
- A \(2 \sqrt{2}\)
- B \(4 \sqrt{2}\)
- C \(\frac{\pi}{4}\)
- D \(\frac{-\pi}{4}\)
Answer & Solution
Correct Answer
(B) \(4 \sqrt{2}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text {} \int_1^{\cos x} t^2 f(t) d t-\cos 2 x \\ & \Rightarrow(\cos x)^2 f(\cos x)\left(\frac{d}{d x} \cos x\right)=-2 \sin 2 x \\ & \Rightarrow \cos ^2 x f(\cos x)(-\sin x)=-2 \sin 2 x \\ & \Rightarrow f(\cos x)=\frac{-2 \sin 2 x}{-\cos ^2 x \sin x} \\ &…
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