Integral of log(sin(x)) dx

The solution

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$$\int\limits_{0}^{\frac{\pi}{2}} \log{\left(\sin{\left(x \right)} \right)}\, dx$$

Integral(log(sin(x)), (x, 0, pi/2))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

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

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

To find $v{\left(x \right)}$ :
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
Now evaluate the sub-integral.
Don't know the steps in finding this integral.

But the integral is

$\int \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx$
Now simplify:

$x \log{\left(\sin{\left(x \right)} \right)} - \int \frac{x}{\tan{\left(x \right)}}\, dx$
Add the constant of integration:

$x \log{\left(\sin{\left(x \right)} \right)} - \int \frac{x}{\tan{\left(x \right)}}\, dx+ \mathrm{constant}$

The answer is:

$x \log{\left(\sin{\left(x \right)} \right)} - \int \frac{x}{\tan{\left(x \right)}}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$\int \log{\left(\sin{\left(x \right)} \right)}\, dx = C + x \log{\left(\sin{\left(x \right)} \right)} - \int \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx$$

Simplify

The answer [src]

 pi               
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 |  log(sin(x)) dx
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0

$$\int\limits_{0}^{\frac{\pi}{2}} \log{\left(\sin{\left(x \right)} \right)}\, dx$$

 pi               
 --               
 2                
  /               
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 |  log(sin(x)) dx
 |                
/                 
0

$$\int\limits_{0}^{\frac{\pi}{2}} \log{\left(\sin{\left(x \right)} \right)}\, dx$$

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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Integral of log(sin(x)) dx

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The solution