1. Tardigrade
  2. Question
  3. Mathematics
  4. If φ(x)=(1/√ x ) ∫ limits(π/4)x(4 √2 sin t-3 φ prime(t)) dt , x>0, then φ prime((π/4)) is equal to :

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. If $\phi(x)=\frac{1}{\sqrt{ x }} \int\limits_{\frac{\pi}{4}}^x\left(4 \sqrt{2} \sin t-3 \phi^{\prime}(t)\right) dt , x>0$, then $\phi^{\prime}\left(\frac{\pi}{4}\right)$ is equal to :

JEE MainJEE Main 2023Integrals

Solution:

$ \phi^{\prime}( x )=\frac{1}{\sqrt{ x }}\left[\left(4 \sqrt{2} \sin x -3 \phi^{\prime}( x )\right) \cdot 1-0\right]-\frac{1}{2} x ^{-3 / 2} $
$ \int\limits_{\frac{\pi}{4}}^{ x }\left(4 \sqrt{2} \sin t -3 \phi^{\prime}( t )\right) dt $
$ \phi^{\prime}\left(\frac{\pi}{4}\right)=\frac{2}{\sqrt{\pi}}\left[4-3 \phi^{\prime}\left(\frac{\pi}{4}\right)\right]+0$
$ \left(1+\frac{6}{\sqrt{\pi}}\right) \phi^{\prime}\left(\frac{\pi}{4}\right)=\frac{8}{\sqrt{\pi}}$
$ \phi^{\prime}\left(\frac{\pi}{4}\right)=\frac{8}{\sqrt{\pi}+6}$