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sin2xdx/sin^4xdx
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Solving integrals/ sin2xdx/sin^4xdx

Integral of sin2xdx/sin^4xdx dx

The solution

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  1                        
  /                        
 |                         
 |                1        
 |  sin(2*x)*1*-------*1 dx
 |                4        
 |             sin (x)     
 |                         
/                          
0

$$\int\limits_{0}^{1} \sin{\left(2 x \right)} 1 \cdot \frac{1}{\sin^{4}{\left(x \right)}} 1\, dx$$

Integral(sin(2*x)*1*1/sin(x)^4, (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{4}{\left(x \right)}}\, dx = 2 \int \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{4}{\left(x \right)}}\, dx$
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int \frac{1}{u^{3}}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{3}}\, du = - \frac{1}{2 u^{2}}$
    Now substitute $u$ back in:
    
    $- \frac{1}{2 \sin^{2}{\left(x \right)}}$
  So, the result is: $- \frac{1}{\sin^{2}{\left(x \right)}}$
Method #2
1. Rewrite the integrand:
  
  $\sin{\left(2 x \right)} 1 \cdot \frac{1}{\sin^{4}{\left(x \right)}} 1 = \frac{2 \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}\, dx = 2 \int \frac{\cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}\, dx$
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int \frac{1}{u^{3}}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{3}}\, du = - \frac{1}{2 u^{2}}$
    Now substitute $u$ back in:
    
    $- \frac{1}{2 \sin^{2}{\left(x \right)}}$
  So, the result is: $- \frac{1}{\sin^{2}{\left(x \right)}}$
Add the constant of integration:

$- \frac{1}{\sin^{2}{\left(x \right)}}+ \mathrm{constant}$

The answer is:

$- \frac{1}{\sin^{2}{\left(x \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                     
 |                                      
 |               1                  1   
 | sin(2*x)*1*-------*1 dx = C - -------
 |               4                  2   
 |            sin (x)            sin (x)
 |                                      
/

$$\int \sin{\left(2 x \right)} 1 \cdot \frac{1}{\sin^{4}{\left(x \right)}} 1\, dx = C - \frac{1}{\sin^{2}{\left(x \right)}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

Упростить

Numerical answer [src]

1.83073007580698e+38

1.83073007580698e+38

The graph

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

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How to use it?

Identical expressions

Integral of sin2xdx/sin^4xdx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2