Question Number 221373 by Nicholas666 last updated on 01/Jun/25
Commented by MathematicalUser2357 last updated on 09/Jun/25
![((((∫_0 ^∞ cos ((3(√3))/4)x^3 dx)/(∫_0 ^∞ sin 16x^3 dx))+(∫_0 ^∞ ln((((1+2x^(2(√2)) +x^(√2) +(√x^(√2) )))/((1+2x^(2(√2)) +x^(√2) −(√x^(√2) )))))(dx/((1+x^2 )ln x))+((21Σ_(n=1) ^∞ arctan (2/n^2 ))/(lim_(t→0^− ) ∫_(−2023) ^(2023) ((tcos x)/(x^2 +t^2 ))dx))))/(lim_(n→∞) (n/2)[(∫_0 ^1 (x^(n−1) /(1+x))dx)−(1/2)])) How???](https://www.tinkutara.com/question/Q221675.png)
$$\frac{\frac{\int_{\mathrm{0}} ^{\infty} \mathrm{cos}\:\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{4}}{x}^{\mathrm{3}} {dx}}{\int_{\mathrm{0}} ^{\infty} \mathrm{sin}\:\mathrm{16}{x}^{\mathrm{3}} {dx}}+\left(\int_{\mathrm{0}} ^{\infty} \mathrm{ln}\left(\frac{\left(\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}\sqrt{\mathrm{2}}} +{x}^{\sqrt{\mathrm{2}}} +\sqrt{{x}^{\sqrt{\mathrm{2}}} }\right)}{\left(\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}\sqrt{\mathrm{2}}} +{x}^{\sqrt{\mathrm{2}}} −\sqrt{{x}^{\sqrt{\mathrm{2}}} }\right)}\right)\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\mathrm{ln}\:{x}}+\frac{\mathrm{21}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{arctan}\:\frac{\mathrm{2}}{{n}^{\mathrm{2}} }}{\underset{{t}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\int_{−\mathrm{2023}} ^{\mathrm{2023}} \frac{{t}\mathrm{cos}\:{x}}{{x}^{\mathrm{2}} +{t}^{\mathrm{2}} }{dx}}\right)}{\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{n}}{\mathrm{2}}\left[\left(\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}−\mathrm{1}} }{\mathrm{1}+{x}}{dx}\right)−\frac{\mathrm{1}}{\mathrm{2}}\right]} \\ $$$$\mathrm{How}??? \\ $$