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Integral of (ctg(x))^3/sin(2x)
Identical expressions
(ctg(x))^ three /sin(2x)
(ctg(x)) cubed divide by sinus of (2x)
(ctg(x)) to the power of three divide by sinus of (2x)
(ctg(x))3/sin(2x)
ctgx3/sin2x
(ctg(x))³/sin(2x)
(ctg(x)) to the power of 3/sin(2x)
ctgx^3/sin2x
(ctg(x))^3 divide by sin(2x)
(ctg(x))^3/sin(2x)dx

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

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

The solution

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

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

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

Detail solution

Rewrite the integrand:

$\frac{\cot^{3}{\left(x \right)}}{\sin{\left(2 x \right)}} = \frac{\cot^{3}{\left(x \right)}}{2 \sin{\left(x \right)} \cos{\left(x \right)}}$
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{\cot^{3}{\left(x \right)}}{2 \sin{\left(x \right)} \cos{\left(x \right)}}\, dx = \frac{\int \frac{\cot^{3}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}}\, dx}{2}$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{\cos{\left(x \right)}}{3 \sin{\left(x \right)}} - \frac{\cos{\left(x \right)}}{3 \sin^{3}{\left(x \right)}}$
So, the result is: $\frac{\cos{\left(x \right)}}{6 \sin{\left(x \right)}} - \frac{\cos{\left(x \right)}}{6 \sin^{3}{\left(x \right)}}$
Now simplify:

$\frac{1}{6 \tan{\left(x \right)}} - \frac{\cos{\left(x \right)}}{6 \sin^{3}{\left(x \right)}}$
Add the constant of integration:

$\frac{1}{6 \tan{\left(x \right)}} - \frac{\cos{\left(x \right)}}{6 \sin^{3}{\left(x \right)}}+ \mathrm{constant}$

The answer is:

$\frac{1}{6 \tan{\left(x \right)}} - \frac{\cos{\left(x \right)}}{6 \sin^{3}{\left(x \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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$$\infty$$

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$$\infty$$

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Numerical answer [src]

3.90715561222928e+56

3.90715561222928e+56

The graph

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution