If (\phi(x)=\frac{1}{\sqrt{ x }} \integral\_{\frac{\pi}{4}}^... | Mathematics | SolveeIt | Solveeit

Today, August 18

Question

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

A

$\frac{4}{6+\sqrt{\pi}}$

B

$\frac{8}{6+\sqrt{\pi}}$

C

$\frac{8}{\sqrt{\pi}}$

D

$\frac{4}{6-\sqrt{\pi}}$

Answer

$\frac{8}{6+\sqrt{\pi}}$

Explanation

Solution

ϕ′(x)=x1[(42sinx−3ϕ′(x))⋅1−0]−21x−3/2
4π∫x(42sint−3ϕ′(t))dt
ϕ′(4π)=π2[4−3ϕ′(4π)]+0
(1+π6)ϕ′(4π)=π8
ϕ′(4π)=π+68
So, thr correct option is (B) : $\frac{8}{6+\sqrt{\pi}}$