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Question Number 113452 by mathdave last updated on 13/Sep/20

Answered by maths mind last updated on 13/Sep/20

$$x=sin\left(t\right)$$$$\Rightarrow ={\int }_0^{\frac{\pi }{2}}\frac{tcos\left(t\right)dt}{sin\left(t\right)+cos\left(t\right)}=I$$$$J={\int }_0^{\frac{\pi }{2}}\frac{tsin\left(t\right)}{sin\left(t\right)+cos\left(t\right)}dt$$$$I+J={\int }_0^{\frac{\pi }{2}}tdt=\frac{{\pi }^2}{8}$$$$I-J={\int }_0^{\frac{\pi }{2}}t\frac{cos\left(t\right)-sin\left(t\right)}{sin\left(t\right)+cos\left(t\right)}dtIBP$$$$={\left[tln(sin\left(t\right)+cos\left(t\right))\right]}_0^{\frac{\pi }{2}}-{\int }_0^{\frac{\pi }{2}}ln(sin\left(t\right)+cos\left(t\right))dt$$$$=-{\int }_0^{\frac{\pi }{2}}ln\left(\sqrt{2c}os(\frac{\pi }{4}-t)\right)dt$$$$=-{\int }_0^{\frac{\pi }{2}}ln\left(\sqrt{2}\right)dt-{\int }_0^{\frac{\pi }{4}}ln\left(cos(\frac{\pi }{4}-t)\right)dt-{\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}ln\left(cos(\frac{\pi }{4}-t)\right)dt$$$$=-\frac{\pi }{4}ln\left(2\right)-{\int }_0^{\frac{\pi }{4}}cos\left(t\right)dt+{\int }_0^{\frac{\pi }{4}}cos(\frac{\pi }{4}-(u+\frac{\pi }{4}))du$$$$=-\frac{\pi }{4}ln\left(2\right)-2{\int }_0^{\frac{\pi }{4}}ln\left(cos\left(t\right)\right)dt$$$$letsfind{\int }_0^{\frac{\pi }{4}}ln\left(cos\left(t\right)\right)d\underset{=A}{t}$$$$weuseG=-{\int }_0^{\frac{\pi }{4}}ln\left(tg\left(t\right)\right)dt=catalanConstante$$$$andcallB={\int }_0^{\frac{\pi }{4}}ln\left(sin\left(t\right)\right)dt$$$$A-B=G$$$$A+B={\int }_0^{\frac{\pi }{4}}ln\left(sin\left(2t\right)/2\right)dt=-ln\left(2\right)\frac{\pi }{4}+\frac{1}{2}{\int }_0^{\frac{\pi }{2}}ln\left(sin\left(x\right)\right)dx$$$$=\frac{-ln\left(2\right)\pi }{4}+\frac{1}{2}.-\frac{\pi log\left(2\right)}{2}=-\frac{\pi log\left(2\right)}{2}$$$$A=\frac{1}{2}(G-\pi \frac{log\left(2\right)}{2})$$$$I-J=-\frac{\pi ln\left(2\right)}{4}-2.\frac{1}{2}(G-\pi log\left(2\right).\frac{1}{2})$$$$=-G+\frac{\pi log\left(2\right)}{4}$$$$I=\frac{1}{2}(-G+\frac{\pi log\left(2\right)}{4}+\frac{{\pi }^2}{8})={\int }_0^1\frac{{\sin }^{-1}\left(x\right)}{x+\sqrt{1-x^2}}dx\backsimeq 0.431$$$$$$$$$$$$$$

Commented by mathdave last updated on 14/Sep/20

$$smilecorrectmankeepthespiritup$$

Commented by Tawa11 last updated on 06/Sep/21

$$\mathrm{great}\mathrm{sir}$$

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