If (x)=1x _()/(4)^x(4 √2 t-3 ^(t)) d t, x0, then ^(()/(4)) is equal to :

Question

If $\phi(x)=\frac{1}{\sqrt{x}} \int\limits_{\frac{\pi}{4}}^x\left(4 \sqrt{2} \sin t-3 \phi^{\prime}(t)\right) d t, x>0$,

then $\emptyset^{\prime}\left(\frac{\pi}{4}\right)$ is equal to :

Accepted Answer

$\phi(x)=\frac{1}{\sqrt{x}} \int_{\pi / 4}^{x}\left(4 \sqrt{2} \sin t-3 \phi^{\prime}(t)ight) d t$

$\Rightarrow \phi^{\prime}(x)=\frac{-1}{2 x^{3 / 2}} \int_{\pi / 4}^{x}\left(4 \sqrt{2} \sin t-3 \phi^{\prime}(t)ight) d t$

$+\frac{1}{\sqrt{x}}\left(4 \sqrt{2} \sin (x)-3 \phi^{\prime}(x)ight)$

At, $x=\frac{\pi}{4}$

$\phi^{\prime}\left(\frac{\pi}{4}ight)=\frac{-1}{2\left(\frac{\pi}{4}ight)^{3 / 2}} 	imes 0+\sqrt{\frac{4}{\pi}}\left(4 \sqrt{2} 	imes \frac{1}{\sqrt{2}}-3 \phi^{\prime}\left(\frac{\pi}{4}ight)ight)$

$\Rightarrow \phi^{\prime}\left(\frac{\pi}{4}ight)\left[1+\frac{6}{\sqrt{\pi}}ight]=\frac{2}{\sqrt{\pi}} 	imes 4$

$\Rightarrow \phi^{\prime}\left(\frac{\pi}{4}ight)=\frac{8}{6+\sqrt{x}}$

Solutions · 1