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Solving integrals/ log(sin(x))

Integral of log(sin(x)) dx

Limits of integration:

from to
v

The graph:

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Piecewise:

The solution

You have entered [src] 
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∫0π2log⁡(sin⁡(x)) dx\int\limits_{0}^{\frac{\pi}{2}} \log{\left(\sin{\left(x \right)} \right)}\, dx 
Integral(log(sin(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)=1\operatorname{dv}{\left(x \right)} = 1.

    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 a constant is the constant times the variable of integration:

      ∫1 dx=x\int 1\, dx = x

    Now evaluate the sub-integral.

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

    But the integral is

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

  3. Now simplify:

    xlog⁡(sin⁡(x))−∫xtan⁡(x) dxx \log{\left(\sin{\left(x \right)} \right)} - \int \frac{x}{\tan{\left(x \right)}}\, dx

  4. Add the constant of integration:

    xlog⁡(sin⁡(x))−∫xtan⁡(x) dx+constantx \log{\left(\sin{\left(x \right)} \right)} - \int \frac{x}{\tan{\left(x \right)}}\, dx+ \mathrm{constant}

The answer is:

xlog⁡(sin⁡(x))−∫xtan⁡(x) dx+constantx \log{\left(\sin{\left(x \right)} \right)} - \int \frac{x}{\tan{\left(x \right)}}\, dx+ \mathrm{constant}

The answer (Indefinite) [src] 
                          /                           
  /                      |                            
 |                       | x*cos(x)                   
 | log(sin(x)) dx = C -  | -------- dx + x*log(sin(x))
 |                       |  sin(x)                    
/                        |                            
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∫log⁡(sin⁡(x)) dx=C+xlog⁡(sin⁡(x))−∫xcos⁡(x)sin⁡(x) dx\int \log{\left(\sin{\left(x \right)} \right)}\, dx = C + x \log{\left(\sin{\left(x \right)} \right)} - \int \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx
Simplify
The answer [src] 
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∫0π2log⁡(sin⁡(x)) dx\int\limits_{0}^{\frac{\pi}{2}} \log{\left(\sin{\left(x \right)} \right)}\, dx 
=
= 
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0
∫0π2log⁡(sin⁡(x)) dx\int\limits_{0}^{\frac{\pi}{2}} \log{\left(\sin{\left(x \right)} \right)}\, dx 
Integral(log(sin(x)), (x, 0, pi/2))
Numerical answer [src] 
-1.0887930451518
-1.0887930451518

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

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