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miv72 [106K]
3 years ago
15

id="TexFormula1" title=" \int\limits^5_1 { \frac{lnR}{ R^{2} } \, dR " alt=" \int\limits^5_1 { \frac{lnR}{ R^{2} } \, dR " align="absmiddle" class="latex-formula">
Mathematics
1 answer:
alex41 [277]3 years ago
5 0
Integrate by parts, setting

\begin{matrix}u=\ln R&&\mathrm dv=\dfrac{\mathrm dR}{R^2}\\\mathrm du=\dfrac{\mathrm dR}R&&v=-\dfrac1R\end{matrix}

So the integral is

\displaystyle\int_1^5\frac{\ln R}{R^2}\,\mathrm dR=-\dfrac{\ln R}R\bigg|_{R=1}^{R=5}+\int_1^5\frac{\mathrm dR}{R^2}
\displaystyle\int_1^5\frac{\ln R}{R^2}\,\mathrm dR=-\left(\frac{\ln5}5-\frac{\ln1}1)-\frac1R\bigg|_{R=1}^{R=5}
\displaystyle\int_1^5\frac{\ln R}{R^2}\,\mathrm dR=-\frac{\ln5}5-\left(\frac15-1\right)
\displaystyle\int_1^5\frac{\ln R}{R^2}\,\mathrm dR=\frac15(4-\ln5)
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