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Integral of d{x}:
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Derivative of:
1/sin^4x
Identical expressions
one /sin^4x
1 divide by sinus of to the power of 4x
one divide by sinus of to the power of 4x
1/sin4x
1/sin⁴x
1 divide by sin^4x
1/sin^4xdx

Integral of 1/sin^4x dx

The solution

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   ___          
 \/ 2           
 -----          
   2            
   /            
  |             
  |      1      
  |   ------- dx
  |      4      
  |   sin (x)   
  |             
 /              
 1

$$\int\limits_{1}^{\frac{\sqrt{2}}{2}} \frac{1}{\sin^{4}{\left(x \right)}}\, dx$$

Integral(1/(sin(x)^4), (x, 1, sqrt(2)/2))

Detail solution

Rewrite the integrand:

$\csc^{4}{\left(x \right)} = \left(\cot^{2}{\left(x \right)} + 1\right) \csc^{2}{\left(x \right)}$
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:
    1. 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: $- \frac{u^{3}}{3}$
    1. 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 $- du$ :
    
    $\int \left(- u^{2}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{2}\, du = - \int u^{2}\, du$
      
      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 $- du$ :
    
    $\int \left(- u^{2}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{2}\, du = - \int u^{2}\, du$
      
      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: $- \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}$
Add the constant of integration:

$- \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}+ \mathrm{constant}$

The answer is:

$- \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                              3   
 |    1                      cot (x)
 | ------- dx = C - cot(x) - -------
 |    4                         3   
 | sin (x)                          
 |                                  
/

$$\int \frac{1}{\sin^{4}{\left(x \right)}}\, dx = C - \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       /  ___\        /  ___\                        
       |\/ 2 |        |\/ 2 |                        
  2*cos|-----|     cos|-----|                        
       \  2  /        \  2  /      cos(1)    2*cos(1)
- ------------ - ------------- + --------- + --------
       /  ___\         /  ___\        3      3*sin(1)
       |\/ 2 |        3|\/ 2 |   3*sin (1)           
  3*sin|-----|   3*sin |-----|                       
       \  2  /         \  2  /

$$- \frac{\cos{\left(\frac{\sqrt{2}}{2} \right)}}{3 \sin^{3}{\left(\frac{\sqrt{2}}{2} \right)}} - \frac{2 \cos{\left(\frac{\sqrt{2}}{2} \right)}}{3 \sin{\left(\frac{\sqrt{2}}{2} \right)}} + \frac{\cos{\left(1 \right)}}{3 \sin^{3}{\left(1 \right)}} + \frac{2 \cos{\left(1 \right)}}{3 \sin{\left(1 \right)}}$$

       /  ___\        /  ___\                        
       |\/ 2 |        |\/ 2 |                        
  2*cos|-----|     cos|-----|                        
       \  2  /        \  2  /      cos(1)    2*cos(1)
- ------------ - ------------- + --------- + --------
       /  ___\         /  ___\        3      3*sin(1)
       |\/ 2 |        3|\/ 2 |   3*sin (1)           
  3*sin|-----|   3*sin |-----|                       
       \  2  /         \  2  /

-2*cos(sqrt(2)/2)/(3*sin(sqrt(2)/2)) - cos(sqrt(2)/2)/(3*sin(sqrt(2)/2)^3) + cos(1)/(3*sin(1)^3) + 2*cos(1)/(3*sin(1))

Numerical answer [src]

-0.974154827019863

-0.974154827019863

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

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How to use it?

Identical expressions

Integral of 1/sin^4x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3