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Q.
Let $f(x)=\frac{4 x^2-3 x+1}{\int\limits_0^{2 \pi x} \sin ^4 t d t}$. If $f^{\prime}\left(\frac{1}{2}\right)=\frac{p}{q \pi}, p, q \in N$, then $(p-q)$ equal
Integrals
Solution:
$f(x)=\frac{4 x^2-4 x+1+x}{\int\limits_0^{2 \pi x} \sin ^2 t d t}=\frac{(2 x-1)^2}{\int\limits_0^{2 \pi x} \sin ^4 t d t}+\frac{x}{\int\limits_0^{2 \pi x} \sin ^4 t d t}$
$f^{\prime}(x)=g^{\prime}(x)+\frac{\int\limits_0^{2 \pi x} \sin ^4 t d t \cdot 1-x \sin ^4(2 \pi x) \cdot 2 \pi}{\left(\int\limits_0^{2 \pi x} \sin ^4 t d t\right)^2}$
$f ^{\prime}\left(\frac{1}{2}\right)= g \cdot\left(\frac{1}{2}\right)+\frac{1}{\int\limits_0^\pi \sin ^4 tdt }=0+\frac{1}{2 \cdot \frac{3 \pi}{16}}=\frac{8}{3 \pi} \Rightarrow p - q =5$