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Solving integrals/ ln(sin(x))/sqrt(x)

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

Limits of integration:

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Piecewise:

The solution

You have entered [src] 
 pi               
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 |  log(sin(x))   
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∫0π2log⁡(sin⁡(x))x dx\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:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\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 n≠−1n \neq -1:

      ∫1x dx=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=2∫xcos⁡(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: 2∫xcos⁡(x)sin⁡(x) dx2 \int \frac{\sqrt{x} \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx

  3. Now simplify:

    2xlog⁡(sin⁡(x))−2∫xtan⁡(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))−2∫xtan⁡(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))−2∫xtan⁡(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                 |                                      
 |                        /                                       
/
(3 sin⁡2(2 x)+3 cos⁡2(2 x)−6 cos⁡(2 x)+3) ∫((40 x2−12 π x+6 log⁡2) sin⁡(2 x)+((6−24 log⁡2) x−3 π) cos⁡(2 x)−6 x+3 π) sin⁡(4 x)+(((24 log⁡2−6) x+3 π) sin⁡(2 x)+(40 x2−12 π x+6 log⁡2) cos⁡(2 x)−6 log⁡2) cos⁡(4 x)+(−80 x2+24 π x−12 log⁡2) sin⁡2(2 x)+((24 log⁡2+6) x−3 π) sin⁡(2 x)+(−80 x2+24 π x−12 log⁡2) cos⁡2(2 x)+(40 x2−12 π x+18 log⁡2) cos⁡(2 x)−6 log⁡26 elog⁡x2 sin⁡2(4 x)−24 elog⁡x2 sin⁡(2 x) sin⁡(4 x)+6 elog⁡x2 cos⁡2(4 x)+(12 elog⁡x2−24 elog⁡x2 cos⁡(2 x)) cos⁡(4 x)+24 elog⁡x2 sin⁡2(2 x)+24 elog⁡x2 cos⁡2(2 x)−24 elog⁡x2 cos⁡(2 x)+6 elog⁡x2  dx+x ((3 sin⁡2(2 x)+3 cos⁡2(2 x)−6 cos⁡(2 x)+3) log⁡(sin⁡2x+cos⁡2x+2 cos⁡x+1)+(3 sin⁡2(2 x)+3 cos⁡2(2 x)−6 cos⁡(2 x)+3) log⁡(sin⁡2x+cos⁡2x−2 cos⁡x+1)−6 log⁡2 sin⁡2(2 x)+(3 π−10 x) sin⁡(2 x)−6 log⁡2 cos⁡2(2 x)+6 log⁡2 cos⁡(2 x))3 sin⁡2(2 x)+3 cos⁡2(2 x)−6 cos⁡(2 x)+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}}
Simplify
The answer [src] 
 pi               
 --               
 2                
  /               
 |                
 |  log(sin(x))   
 |  ----------- dx
 |       ___      
 |     \/ x       
 |                
/                 
0
∫0π2log⁡(sin⁡(x))x dx\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))x dx\int\limits_{0}^{\frac{\pi}{2}} \frac{\log{\left(\sin{\left(x \right)} \right)}}{\sqrt{x}}\, dx
Упростить
Numerical answer [src] 
-4.0980831368609
-4.0980831368609

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

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