Интеграл x*log(sin(x)) (dx)

Вы ввели

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 |  x*log(sin(x)) dx
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$$\int_{0}^{1} x \log{\left (\sin{\left (x \right )} \right )}\, dx$$

Ответ

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 |  x*log(sin(x)) dx =  |  x*log(sin(x)) dx
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$$-{{i\,\arctan \left({{\sin 1}\over{\cos 1+1}}\right)}\over{2}}-{{i \,\arctan \left({{\sin 1}\over{\cos 1-1}}\right)}\over{2}}+{{\log \sin 1}\over{2}}-{{\log \left(2\,\cos 1+2\right)}\over{4}}-{{\log \left(2-2\,\cos 1\right)}\over{4}}-{\it li}_{3}(e^{i})+i\,{\it li}_{ 2}(e^{i})-{\it li}_{3}(-e^{i})+i\,{\it li}_{2}(-e^{i})+{{\zeta\left( 3\right)}\over{4}}+{{i}\over{6}}$$

Численный ответ

-0.292639521900471

Ответ (Неопределённый)

                            /                             
                           |                              
                           |  2                           
                           | x *cos(x)                    
                           | --------- dx                 
                           |   sin(x)                     
  /                        |                 2            
 |                        /                 x *log(sin(x))
 | x*log(sin(x)) dx = C - --------------- + --------------
 |                               2                2       
/                                                         

$${{x^2\,\log \sin x}\over{2}}-{{3\,x^2\,\log \left(\sin ^2x+\cos ^2x +2\,\cos x+1\right)+3\,x^2\,\log \left(\sin ^2x+\cos ^2x-2\,\cos x+1 \right)+6\,i\,x^2\,{\rm atan2}\left(\sin x , \cos x+1\right)-6\,i\,x ^2\,{\rm atan2}\left(\sin x , 1-\cos x\right)+12\,{\it li}_{3}(e^{i \,x})-12\,i\,x\,{\it li}_{2}(e^{i\,x})+12\,{\it li}_{3}(-e^{i\,x})- 12\,i\,x\,{\it li}_{2}(-e^{i\,x})-2\,i\,x^3}\over{12}}$$