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

The solution

You have entered [src]

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

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

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

Detail solution

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 $\operatorname{dv}{\left(x \right)} = \frac{1}{\sqrt{x}}$ .

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 $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int \frac{1}{\sqrt{x}}\, dx = 2 \sqrt{x}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{2 \sqrt{x} \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx = 2 \int \frac{\sqrt{x} \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\int \frac{\sqrt{x} \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx$
So, the result is: $2 \int \frac{\sqrt{x} \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx$
Now simplify:

$2 \sqrt{x} \log{\left(\sin{\left(x \right)} \right)} - 2 \int \frac{\sqrt{x}}{\tan{\left(x \right)}}\, dx$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$${{\left(3\,\sin ^2\left(2\,x\right)+3\,\cos ^2\left(2\,x\right)-6\, \cos \left(2\,x\right)+3\right)\,\int {{{\left(\left(40\,x^2-12\,\pi \,x+6\,\log 2\right)\,\sin \left(2\,x\right)+\left(\left(6-24\,\log 2\right)\,x-3\,\pi\right)\,\cos \left(2\,x\right)-6\,x+3\,\pi\right) \,\sin \left(4\,x\right)+\left(\left(\left(24\,\log 2-6\right)\,x+3 \,\pi\right)\,\sin \left(2\,x\right)+\left(40\,x^2-12\,\pi\,x+6\, \log 2\right)\,\cos \left(2\,x\right)-6\,\log 2\right)\,\cos \left(4 \,x\right)+\left(-80\,x^2+24\,\pi\,x-12\,\log 2\right)\,\sin ^2 \left(2\,x\right)+\left(\left(24\,\log 2+6\right)\,x-3\,\pi\right)\, \sin \left(2\,x\right)+\left(-80\,x^2+24\,\pi\,x-12\,\log 2\right)\, \cos ^2\left(2\,x\right)+\left(40\,x^2-12\,\pi\,x+18\,\log 2\right) \,\cos \left(2\,x\right)-6\,\log 2}\over{6\,e^{{{\log x}\over{2}}}\, \sin ^2\left(4\,x\right)-24\,e^{{{\log x}\over{2}}}\,\sin \left(2\,x \right)\,\sin \left(4\,x\right)+6\,e^{{{\log x}\over{2}}}\,\cos ^2 \left(4\,x\right)+\left(12\,e^{{{\log x}\over{2}}}-24\,e^{{{\log x }\over{2}}}\,\cos \left(2\,x\right)\right)\,\cos \left(4\,x\right)+ 24\,e^{{{\log x}\over{2}}}\,\sin ^2\left(2\,x\right)+24\,e^{{{\log x }\over{2}}}\,\cos ^2\left(2\,x\right)-24\,e^{{{\log x}\over{2}}}\, \cos \left(2\,x\right)+6\,e^{{{\log x}\over{2}}}}}}{\;dx}+\sqrt{x}\, \left(\left(3\,\sin ^2\left(2\,x\right)+3\,\cos ^2\left(2\,x\right)- 6\,\cos \left(2\,x\right)+3\right)\,\log \left(\sin ^2x+\cos ^2x+2\, \cos x+1\right)+\left(3\,\sin ^2\left(2\,x\right)+3\,\cos ^2\left(2 \,x\right)-6\,\cos \left(2\,x\right)+3\right)\,\log \left(\sin ^2x+ \cos ^2x-2\,\cos x+1\right)-6\,\log 2\,\sin ^2\left(2\,x\right)+ \left(3\,\pi-10\,x\right)\,\sin \left(2\,x\right)-6\,\log 2\,\cos ^2 \left(2\,x\right)+6\,\log 2\,\cos \left(2\,x\right)\right)}\over{3\, \sin ^2\left(2\,x\right)+3\,\cos ^2\left(2\,x\right)-6\,\cos \left(2 \,x\right)+3}}$$

The answer [src]

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

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

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

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

Упростить

Numerical answer [src]

-4.0980831368609

-4.0980831368609

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

Other calculators

How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution