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Question Number 99261 by Ar Brandon last updated on 19/Jun/20

$$\mathrm{Given}\mathrm{F}\left(\mathrm{x}\right)=\frac{1}{2}\int \frac{\mathrm{x}+1}{\mathrm{x}-1}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}$$$$\mathrm{Show}\mathrm{that}\mathrm{F}\mathrm{is}\mathrm{defined},\mathrm{continuous},\mathrm{and}\mathrm{derivable}.$$$$\mathrm{And}\mathrm{find}\mathrm{its}\mathrm{derivative}$$

Answered by abdomathmax last updated on 19/Jun/20

$$\mathrm{F}\left(\mathrm{x}\right)=\frac{1}{2}\frac{\mathrm{x}+1}{\mathrm{x}-1}{\int }^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}\Rightarrow {\mathrm{F}}^{{}^{\prime }}\left(\mathrm{x}\right)=\frac{1}{2}{\left(\frac{\mathrm{x}+1}{\mathrm{x}-1}\right)}^{{}^{\prime }}{\int }^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}$$$$+\frac{\mathrm{x}+1}{2(\mathrm{x}-1)}{\mathrm{f}}^{{}^{\prime }}\left(\mathrm{x}\right)=\frac{1}{2}\times \frac{\mathrm{x}-1-(\mathrm{x}+1)}{{(\mathrm{x}-1)}^2}{\int }^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}$$$$+\frac{\mathrm{x}+1}{2(\mathrm{x}-1)}{\mathrm{f}}^{{}^{\prime }}\left(\mathrm{x}\right)=-\frac{1}{{(\mathrm{x}-1)}^2}{\int }^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}+\frac{\mathrm{x}+1}{2(\mathrm{x}-1)}{\mathrm{f}}^{{}^{\prime }}\left(\mathrm{x}\right)$$

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