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Integral of ln(sin(x))/sqrt(x) dx

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The solution

You have entered [src]
 pi               
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  /               
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 |  log(sin(x))   
 |  ----------- dx
 |       ___      
 |     \/ x       
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0                 
0π2log(sin(x))xdx\int\limits_{0}^{\frac{\pi}{2}} \frac{\log{\left(\sin{\left(x \right)} \right)}}{\sqrt{x}}\, dx
Integral(log(sin(x))/(sqrt(x)), (x, 0, pi/2))
Detail solution
  1. Use integration by parts:

    udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=log(sin(x))u{\left(x \right)} = \log{\left(\sin{\left(x \right)} \right)} and let dv(x)=1x\operatorname{dv}{\left(x \right)} = \frac{1}{\sqrt{x}}.

    Then du(x)=cos(x)sin(x)\operatorname{du}{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}.

    To find v(x)v{\left(x \right)}:

    1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

      1xdx=2x\int \frac{1}{\sqrt{x}}\, dx = 2 \sqrt{x}

    Now evaluate the sub-integral.

  2. The integral of a constant times a function is the constant times the integral of the function:

    2xcos(x)sin(x)dx=2xcos(x)sin(x)dx\int \frac{2 \sqrt{x} \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx = 2 \int \frac{\sqrt{x} \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx

    1. Don't know the steps in finding this integral.

      But the integral is

      xcos(x)sin(x)dx\int \frac{\sqrt{x} \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx

    So, the result is: 2xcos(x)sin(x)dx2 \int \frac{\sqrt{x} \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx

  3. Now simplify:

    2xlog(sin(x))2xtan(x)dx2 \sqrt{x} \log{\left(\sin{\left(x \right)} \right)} - 2 \int \frac{\sqrt{x}}{\tan{\left(x \right)}}\, dx

  4. Add the constant of integration:

    2xlog(sin(x))2xtan(x)dx+constant2 \sqrt{x} \log{\left(\sin{\left(x \right)} \right)} - 2 \int \frac{\sqrt{x}}{\tan{\left(x \right)}}\, dx+ \mathrm{constant}


The answer is:

2xlog(sin(x))2xtan(x)dx+constant2 \sqrt{x} \log{\left(\sin{\left(x \right)} \right)} - 2 \int \frac{\sqrt{x}}{\tan{\left(x \right)}}\, dx+ \mathrm{constant}

The answer (Indefinite) [src]
                            /                                     
  /                        |                                      
 |                         |   ___                                
 | log(sin(x))             | \/ x *cos(x)          ___            
 | ----------- dx = C - 2* | ------------ dx + 2*\/ x *log(sin(x))
 |      ___                |    sin(x)                            
 |    \/ x                 |                                      
 |                        /                                       
/                                                                 
(3sin2(2x)+3cos2(2x)6cos(2x)+3)((40x212πx+6log2)sin(2x)+((624log2)x3π)cos(2x)6x+3π)sin(4x)+(((24log26)x+3π)sin(2x)+(40x212πx+6log2)cos(2x)6log2)cos(4x)+(80x2+24πx12log2)sin2(2x)+((24log2+6)x3π)sin(2x)+(80x2+24πx12log2)cos2(2x)+(40x212πx+18log2)cos(2x)6log26elogx2sin2(4x)24elogx2sin(2x)sin(4x)+6elogx2cos2(4x)+(12elogx224elogx2cos(2x))cos(4x)+24elogx2sin2(2x)+24elogx2cos2(2x)24elogx2cos(2x)+6elogx2  dx+x((3sin2(2x)+3cos2(2x)6cos(2x)+3)log(sin2x+cos2x+2cosx+1)+(3sin2(2x)+3cos2(2x)6cos(2x)+3)log(sin2x+cos2x2cosx+1)6log2sin2(2x)+(3π10x)sin(2x)6log2cos2(2x)+6log2cos(2x))3sin2(2x)+3cos2(2x)6cos(2x)+3{{\left(3\,\sin ^2\left(2\,x\right)+3\,\cos ^2\left(2\,x\right)-6\, \cos \left(2\,x\right)+3\right)\,\int {{{\left(\left(40\,x^2-12\,\pi \,x+6\,\log 2\right)\,\sin \left(2\,x\right)+\left(\left(6-24\,\log 2\right)\,x-3\,\pi\right)\,\cos \left(2\,x\right)-6\,x+3\,\pi\right) \,\sin \left(4\,x\right)+\left(\left(\left(24\,\log 2-6\right)\,x+3 \,\pi\right)\,\sin \left(2\,x\right)+\left(40\,x^2-12\,\pi\,x+6\, \log 2\right)\,\cos \left(2\,x\right)-6\,\log 2\right)\,\cos \left(4 \,x\right)+\left(-80\,x^2+24\,\pi\,x-12\,\log 2\right)\,\sin ^2 \left(2\,x\right)+\left(\left(24\,\log 2+6\right)\,x-3\,\pi\right)\, \sin \left(2\,x\right)+\left(-80\,x^2+24\,\pi\,x-12\,\log 2\right)\, \cos ^2\left(2\,x\right)+\left(40\,x^2-12\,\pi\,x+18\,\log 2\right) \,\cos \left(2\,x\right)-6\,\log 2}\over{6\,e^{{{\log x}\over{2}}}\, \sin ^2\left(4\,x\right)-24\,e^{{{\log x}\over{2}}}\,\sin \left(2\,x \right)\,\sin \left(4\,x\right)+6\,e^{{{\log x}\over{2}}}\,\cos ^2 \left(4\,x\right)+\left(12\,e^{{{\log x}\over{2}}}-24\,e^{{{\log x }\over{2}}}\,\cos \left(2\,x\right)\right)\,\cos \left(4\,x\right)+ 24\,e^{{{\log x}\over{2}}}\,\sin ^2\left(2\,x\right)+24\,e^{{{\log x }\over{2}}}\,\cos ^2\left(2\,x\right)-24\,e^{{{\log x}\over{2}}}\, \cos \left(2\,x\right)+6\,e^{{{\log x}\over{2}}}}}}{\;dx}+\sqrt{x}\, \left(\left(3\,\sin ^2\left(2\,x\right)+3\,\cos ^2\left(2\,x\right)- 6\,\cos \left(2\,x\right)+3\right)\,\log \left(\sin ^2x+\cos ^2x+2\, \cos x+1\right)+\left(3\,\sin ^2\left(2\,x\right)+3\,\cos ^2\left(2 \,x\right)-6\,\cos \left(2\,x\right)+3\right)\,\log \left(\sin ^2x+ \cos ^2x-2\,\cos x+1\right)-6\,\log 2\,\sin ^2\left(2\,x\right)+ \left(3\,\pi-10\,x\right)\,\sin \left(2\,x\right)-6\,\log 2\,\cos ^2 \left(2\,x\right)+6\,\log 2\,\cos \left(2\,x\right)\right)}\over{3\, \sin ^2\left(2\,x\right)+3\,\cos ^2\left(2\,x\right)-6\,\cos \left(2 \,x\right)+3}}
The answer [src]
 pi               
 --               
 2                
  /               
 |                
 |  log(sin(x))   
 |  ----------- dx
 |       ___      
 |     \/ x       
 |                
/                 
0                 
0π2log(sin(x))xdx\int\limits_{0}^{\frac{\pi}{2}} \frac{\log{\left(\sin{\left(x \right)} \right)}}{\sqrt{x}}\, dx
=
=
 pi               
 --               
 2                
  /               
 |                
 |  log(sin(x))   
 |  ----------- dx
 |       ___      
 |     \/ x       
 |                
/                 
0                 
0π2log(sin(x))xdx\int\limits_{0}^{\frac{\pi}{2}} \frac{\log{\left(\sin{\left(x \right)} \right)}}{\sqrt{x}}\, dx
Numerical answer [src]
-4.0980831368609
-4.0980831368609

    Use the examples entering the upper and lower limits of integration.