Question $1$

Let $F(x)={\displaystyle \underset{1}{\overset{cosx}{}}}\frac{1}{1+t^7}dt$ for $xin(-,).$ Find $F^{\text{'}}(x).$

$F^{\text{'}}(x)=\frac{1}{1+co{s}^{7}x}$

$F^{\text{'}}(x)=\frac{-sinx}{1+co{s}^{7}x}$

$F^{\text{'}}(x)=\frac{-8sinxcosx}{8cosx+co{s}^{8}x}$

$F^{\text{'}}(x)=\frac{1}{1-si{n}^{7}x}$

$F^{\text{'}}(x)=\frac{8cosx}{8cosx+co{s}^{8}x}$