Integral of x/sin²x² dx

The solution

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  1                
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0

$$\int\limits_{0}^{1} \frac{x}{\sin^{2^{2}}{\left(x \right)}}\, dx$$

Integral(x/(sin(x)^(2^2)), (x, 0, 1))

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \frac{1}{\sin^{4}{\left(x \right)}}$ .

Then $\operatorname{du}{\left(x \right)} = 1$ .

To find $v{\left(x \right)}$ :
1. Rewrite the integrand:
  
  $\csc^{4}{\left(x \right)} = \left(\cot^{2}{\left(x \right)} + 1\right) \csc^{2}{\left(x \right)}$
2. There are multiple ways to do this integral.
  Method #1
  1. Let $u = \cot{\left(x \right)}$ .
    
    Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ and substitute $du$ :
    
    $\int \left(- u^{2} - 1\right)\, du$
    1. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $- \frac{u^{3}}{3}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \left(-1\right)\, du = - u$
      
      The result is: $- \frac{u^{3}}{3} - u$
    Now substitute $u$ back in:
    
    $- \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}$
  Method #2
  1. Rewrite the integrand:
    
    $\left(\cot^{2}{\left(x \right)} + 1\right) \csc^{2}{\left(x \right)} = \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)} + \csc^{2}{\left(x \right)}$
  2. Integrate term-by-term:
    1. Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ and substitute $- du$ :
      
      $\int u^{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $- \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cot^{3}{\left(x \right)}}{3}$
    1. $\int \csc^{2}{\left(x \right)}\, dx = - \cot{\left(x \right)}$
    The result is: $- \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}$
  Method #3
  1. Rewrite the integrand:
    
    $\left(\cot^{2}{\left(x \right)} + 1\right) \csc^{2}{\left(x \right)} = \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)} + \csc^{2}{\left(x \right)}$
  2. Integrate term-by-term:
    1. Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ and substitute $- du$ :
      
      $\int u^{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $- \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cot^{3}{\left(x \right)}}{3}$
    1. $\int \csc^{2}{\left(x \right)}\, dx = - \cot{\left(x \right)}$
    The result is: $- \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}$
Now evaluate the sub-integral.
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\cot^{3}{\left(x \right)}}{3}\right)\, dx = - \frac{\int \cot^{3}{\left(x \right)}\, dx}{3}$
  1. Rewrite the integrand:
    
    $\cot^{3}{\left(x \right)} = \left(\csc^{2}{\left(x \right)} - 1\right) \cot{\left(x \right)}$
  2. Let $u = \csc^{2}{\left(x \right)}$ .
    
    Then let $du = - 2 \cot{\left(x \right)} \csc^{2}{\left(x \right)} dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{1 - u}{4 u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1 - u}{2 u}\, du = \frac{\int \frac{1 - u}{u}\, du}{2}$
      1. Let $u = - u$ .
        
        Then let $du = - du$ and substitute $du$ :
        
        $\int \frac{u + 1}{u}\, du$
        
        Rewrite the integrand:
        
        $\frac{u + 1}{u} = 1 + \frac{1}{u}$
        
        Integrate term-by-term:
        
        The integral of a constant is the constant times the variable of integration:
        
        $\int 1\, du = u$
        
        The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
        
        The result is: $u + \log{\left(u \right)}$
        
        Now substitute $u$ back in:
        
        $- u + \log{\left(- u \right)}$
      So, the result is: $- \frac{u}{2} + \frac{\log{\left(- u \right)}}{2}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(- \csc^{2}{\left(x \right)} \right)}}{2} - \frac{\csc^{2}{\left(x \right)}}{2}$
  So, the result is: $- \frac{\log{\left(- \csc^{2}{\left(x \right)} \right)}}{6} + \frac{\csc^{2}{\left(x \right)}}{6}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \cot{\left(x \right)}\right)\, dx = - \int \cot{\left(x \right)}\, dx$
  1. Rewrite the integrand:
    
    $\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
  2. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int \frac{1}{u}\, du$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    Now substitute $u$ back in:
    
    $\log{\left(\sin{\left(x \right)} \right)}$
  So, the result is: $- \log{\left(\sin{\left(x \right)} \right)}$
The result is: $- \log{\left(\sin{\left(x \right)} \right)} - \frac{\log{\left(- \csc^{2}{\left(x \right)} \right)}}{6} + \frac{\csc^{2}{\left(x \right)}}{6}$
Now simplify:

$- \frac{x \left(\cot^{2}{\left(x \right)} + 3\right) \cot{\left(x \right)}}{3} + \log{\left(\sin{\left(x \right)} \right)} + \frac{\log{\left(- \csc^{2}{\left(x \right)} \right)}}{6} - \frac{\csc^{2}{\left(x \right)}}{6}$
Add the constant of integration:

$- \frac{x \left(\cot^{2}{\left(x \right)} + 3\right) \cot{\left(x \right)}}{3} + \log{\left(\sin{\left(x \right)} \right)} + \frac{\log{\left(- \csc^{2}{\left(x \right)} \right)}}{6} - \frac{\csc^{2}{\left(x \right)}}{6}+ \mathrm{constant}$

The answer is:

$- \frac{x \left(\cot^{2}{\left(x \right)} + 3\right) \cot{\left(x \right)}}{3} + \log{\left(\sin{\left(x \right)} \right)} + \frac{\log{\left(- \csc^{2}{\left(x \right)} \right)}}{6} - \frac{\csc^{2}{\left(x \right)}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                   
 |                          2         /    2   \     /             3   \              
 |      x                csc (x)   log\-csc (x)/     |          cot (x)|              
 | ------------ dx = C - ------- + ------------- + x*|-cot(x) - -------| + log(sin(x))
 |         / 2\             6            6           \             3   /              
 |         \2 /                                                                       
 | (sin(x))                                                                           
 |                                                                                    
/

$$\int \frac{x}{\sin^{2^{2}}{\left(x \right)}}\, dx = C + x \left(- \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}\right) + \log{\left(\sin{\left(x \right)} \right)} + \frac{\log{\left(- \csc^{2}{\left(x \right)} \right)}}{6} - \frac{\csc^{2}{\left(x \right)}}{6}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

Упростить

Numerical answer [src]

9.15365037903492e+37

9.15365037903492e+37

The graph

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

Other calculators

How to use it?

Identical expressions

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Integral of x/sin²x² dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3