0-e-s-s-5-ds-2-e-t-2-2-dt-0-sint-2-dt-n-0-1-n-2n-1-0-sin-x-x-dx-n-1-arctan-2-n-2-lim-t-0-2020-

Question Number 212416 by MrGaster last updated on 13/Oct/24

$$\frac{\frac{\int_{\mathrm{0}} ^{+\infty} {e}^{−{s}} {s}^{\mathrm{5}} {ds}}{\mathrm{2}}+\frac{\int_{−\infty} ^{+\infty} {e}^{−\frac{{t}^{\mathrm{2}} }{\mathrm{2}}} {dt}}{\int_{\mathrm{0}} ^{+\infty} \mathrm{sin}{t}^{\mathrm{2}} {dt}}\left(\frac{\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{2}{n}+\mathrm{1}}}{\int_{\mathrm{0}} ^{+\infty} \frac{\mathrm{sin}\:{x}}{{x}}{dx}}+\frac{\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{arctan}\frac{\mathrm{2}}{{n}^{\mathrm{2}} }}{\underset{{t}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\int_{−\mathrm{2020}} ^{\mathrm{2020}} \frac{{t}\mathrm{cos}\:{x}}{{x}^{\mathrm{2}} +{t}^{\mathrm{2}} }{dx}}\right)}{\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left\{\left[\left(\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}−\mathrm{1}} }{\mathrm{1}+{x}}{dx}\right){n}−\frac{\mathrm{1}}{\mathrm{2}}\right]\frac{{n}}{\mathrm{2}}\right\}} \\ $$

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