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Solving integrals/ (3+(2ctg^3(x)))/sin^2(x)

Integral of (3+(2ctg^3(x)))/sin^2(x) dx

The solution

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  1                 
  /                 
 |                  
 |           3      
 |  3 + 2*cot (x)   
 |  ------------- dx
 |        2         
 |     sin (x)      
 |                  
/                   
0

$$\int\limits_{0}^{1} \frac{2 \cot^{3}{\left(x \right)} + 3}{\sin^{2}{\left(x \right)}}\, dx$$

Integral((3 + 2*cot(x)^3)/(sin(x)^2), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{2 \cot^{3}{\left(x \right)} + 3}{\sin^{2}{\left(x \right)}} = \frac{2 \cot^{3}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{3}{\sin^{2}{\left(x \right)}}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 \cot^{3}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\, dx = 2 \int \frac{\cot^{3}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\, dx$
  1. Rewrite the integrand:
    
    $\cot^{3}{\left(x \right)} \csc^{2}{\left(x \right)} = \left(\csc^{2}{\left(x \right)} - 1\right) \cot{\left(x \right)} \csc^{2}{\left(x \right)}$
  2. There are multiple ways to do this integral.
    Method #1
    1. Let $u = \csc^{2}{\left(x \right)}$ .
      
      Then let $du = - 2 \cot{\left(x \right)} \csc^{2}{\left(x \right)} dx$ and substitute $du$ :
      
      $\int \left(\frac{1}{2} - \frac{u}{2}\right)\, du$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, du = \frac{u}{2}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{u}{2}\right)\, du = - \frac{\int u\, du}{2}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{u^{2}}{4}$
      
      The result is: $- \frac{u^{2}}{4} + \frac{u}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\csc^{4}{\left(x \right)}}{4} + \frac{\csc^{2}{\left(x \right)}}{2}$
    Method #2
    1. Rewrite the integrand:
      
      $\left(\csc^{2}{\left(x \right)} - 1\right) \cot{\left(x \right)} \csc^{2}{\left(x \right)} = \cot{\left(x \right)} \csc^{4}{\left(x \right)} - \cot{\left(x \right)} \csc^{2}{\left(x \right)}$
    2. Integrate term-by-term:
      
      Let $u = \csc^{4}{\left(x \right)}$ .
      
      Then let $du = - 4 \cot{\left(x \right)} \csc^{4}{\left(x \right)} dx$ and substitute $- \frac{du}{4}$ :
      
      $\int \frac{1}{16}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{4}\right)\, du = - \frac{\int 1\, du}{4}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $- \frac{u}{4}$
      
      Now substitute $u$ back in:
      
      $- \frac{\csc^{4}{\left(x \right)}}{4}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \cot{\left(x \right)} \csc^{2}{\left(x \right)}\right)\, dx = - \int \cot{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \csc^{2}{\left(x \right)}$ .
      
      Then let $du = - 2 \cot{\left(x \right)} \csc^{2}{\left(x \right)} dx$ and substitute $- \frac{du}{2}$ :
      
      $\int \frac{1}{4}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{2}\right)\, du = - \frac{\int 1\, du}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $- \frac{u}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\csc^{2}{\left(x \right)}}{2}$
      
      So, the result is: $\frac{\csc^{2}{\left(x \right)}}{2}$
      
      The result is: $- \frac{\csc^{4}{\left(x \right)}}{4} + \frac{\csc^{2}{\left(x \right)}}{2}$
    Method #3
    1. Rewrite the integrand:
      
      $\left(\csc^{2}{\left(x \right)} - 1\right) \cot{\left(x \right)} \csc^{2}{\left(x \right)} = \cot{\left(x \right)} \csc^{4}{\left(x \right)} - \cot{\left(x \right)} \csc^{2}{\left(x \right)}$
    2. Integrate term-by-term:
      
      Let $u = \csc^{4}{\left(x \right)}$ .
      
      Then let $du = - 4 \cot{\left(x \right)} \csc^{4}{\left(x \right)} dx$ and substitute $- \frac{du}{4}$ :
      
      $\int \frac{1}{16}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{4}\right)\, du = - \frac{\int 1\, du}{4}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $- \frac{u}{4}$
      
      Now substitute $u$ back in:
      
      $- \frac{\csc^{4}{\left(x \right)}}{4}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \cot{\left(x \right)} \csc^{2}{\left(x \right)}\right)\, dx = - \int \cot{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \csc^{2}{\left(x \right)}$ .
      
      Then let $du = - 2 \cot{\left(x \right)} \csc^{2}{\left(x \right)} dx$ and substitute $- \frac{du}{2}$ :
      
      $\int \frac{1}{4}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{2}\right)\, du = - \frac{\int 1\, du}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $- \frac{u}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\csc^{2}{\left(x \right)}}{2}$
      
      So, the result is: $\frac{\csc^{2}{\left(x \right)}}{2}$
      
      The result is: $- \frac{\csc^{4}{\left(x \right)}}{4} + \frac{\csc^{2}{\left(x \right)}}{2}$
  So, the result is: $- \frac{\csc^{4}{\left(x \right)}}{2} + \csc^{2}{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{3}{\sin^{2}{\left(x \right)}}\, dx = 3 \int \frac{1}{\sin^{2}{\left(x \right)}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $- \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
  So, the result is: $- \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}}$
The result is: $- \frac{\csc^{4}{\left(x \right)}}{2} + \csc^{2}{\left(x \right)} - \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}}$
Now simplify:

$- \frac{3}{\tan{\left(x \right)}} + \frac{1}{\sin^{2}{\left(x \right)}} - \frac{1}{2 \sin^{4}{\left(x \right)}}$
Add the constant of integration:

$- \frac{3}{\tan{\left(x \right)}} + \frac{1}{\sin^{2}{\left(x \right)}} - \frac{1}{2 \sin^{4}{\left(x \right)}}+ \mathrm{constant}$

The answer is:

$- \frac{3}{\tan{\left(x \right)}} + \frac{1}{\sin^{2}{\left(x \right)}} - \frac{1}{2 \sin^{4}{\left(x \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                   
 |                                                    
 |          3                          4              
 | 3 + 2*cot (x)             2      csc (x)   3*cos(x)
 | ------------- dx = C + csc (x) - ------- - --------
 |       2                             2       sin(x) 
 |    sin (x)                                         
 |                                                    
/

$$-{{6\,\tan ^3x+1}\over{2\,\tan ^4x}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$${\it \%a}$$

oo

$$\infty$$

Упростить

Numerical answer [src]

1.45349812331627e+76

1.45349812331627e+76

The graph

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

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Identical expressions

Similar expressions

Integral of (3+(2ctg^3(x)))/sin^2(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3