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log(sinx^3)

Integral of log(sin(x)^3) dx

The solution

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  1                
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 |     /   3   \   
 |  log\sin (x)/ dx
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/                  
0

$$\int\limits_{0}^{1} \log{\left(\sin^{3}{\left(x \right)} \right)}\, dx$$

Integral(log(sin(x)^3), (x, 0, 1))

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = \log{\left(\sin^{3}{\left(x \right)} \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .

Then $\operatorname{du}{\left(x \right)} = \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}}$ .

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

$\int \frac{3 x \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx = 3 \int \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\int \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx$
So, the result is: $3 \int \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx$
Now simplify:

$x \log{\left(\sin^{3}{\left(x \right)} \right)} - 3 \int \frac{x}{\tan{\left(x \right)}}\, dx$
Add the constant of integration:

$x \log{\left(\sin^{3}{\left(x \right)} \right)} - 3 \int \frac{x}{\tan{\left(x \right)}}\, dx+ \mathrm{constant}$

The answer is:

$x \log{\left(\sin^{3}{\left(x \right)} \right)} - 3 \int \frac{x}{\tan{\left(x \right)}}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                          /                            
 |                          |                             
 |    /   3   \             | x*cos(x)           /   3   \
 | log\sin (x)/ dx = C - 3* | -------- dx + x*log\sin (x)/
 |                          |  sin(x)                     
/                           |                             
                           /

$$\int \log{\left(\sin^{3}{\left(x \right)} \right)}\, dx = C + x \log{\left(\sin^{3}{\left(x \right)} \right)} - 3 \int \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx$$

Simplify

The answer [src]

  1                
  /                
 |                 
 |     /   3   \   
 |  log\sin (x)/ dx
 |                 
/                  
0

$$\int\limits_{0}^{1} \log{\left(\sin^{3}{\left(x \right)} \right)}\, dx$$

  1                
  /                
 |                 
 |     /   3   \   
 |  log\sin (x)/ dx
 |                 
/                  
0

$$\int\limits_{0}^{1} \log{\left(\sin^{3}{\left(x \right)} \right)}\, dx$$

Integral(log(sin(x)^3), (x, 0, 1))

Numerical answer [src]

-3.17016061797475

-3.17016061797475

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

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Integral of log(sin(x)^3) dx

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The solution