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  • Integral of d{x}:
  • Integral of e^(x/2) Integral of e^(x/2)
  • Integral of e^(3*x)*dx Integral of e^(3*x)*dx
  • Integral of x^2*exp(x^3) Integral of x^2*exp(x^3)
  • Integral of dy/y^2 Integral of dy/y^2
  • Identical expressions

  • ctg^ five (x)/sin^ two (x)
  • ctg to the power of 5(x) divide by sinus of squared (x)
  • ctg to the power of five (x) divide by sinus of to the power of two (x)
  • ctg5(x)/sin2(x)
  • ctg5x/sin2x
  • ctg⁵(x)/sin²(x)
  • ctg to the power of 5(x)/sin to the power of 2(x)
  • ctg^5x/sin^2x
  • ctg^5(x) divide by sin^2(x)
  • ctg^5(x)/sin^2(x)dx

Integral of ctg^5(x)/sin^2(x) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
 pi           
 --           
 3            
  /           
 |            
 |     5      
 |  cot (x)   
 |  ------- dx
 |     2      
 |  sin (x)   
 |            
/             
pi            
--            
4             
$$\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{\cot^{5}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\, dx$$
Integral(cot(x)^5/sin(x)^2, (x, pi/4, pi/3))
Detail solution
  1. Rewrite the integrand:

  2. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      1. The integral of a constant times a function is the constant times the integral of the function:

        1. The integral of is when :

        So, the result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Let .

        Then let and substitute :

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. The integral of is when :

          So, the result is:

        Now substitute back in:

      1. The integral of a constant times a function is the constant times the integral of the function:

        1. Let .

          Then let and substitute :

          1. The integral of a constant times a function is the constant times the integral of the function:

            1. The integral of is when :

            So, the result is:

          Now substitute back in:

        So, the result is:

      1. Let .

        Then let and substitute :

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. The integral of is when :

          So, the result is:

        Now substitute back in:

      The result is:

    Method #3

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Let .

        Then let and substitute :

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. The integral of is when :

          So, the result is:

        Now substitute back in:

      1. The integral of a constant times a function is the constant times the integral of the function:

        1. Let .

          Then let and substitute :

          1. The integral of a constant times a function is the constant times the integral of the function:

            1. The integral of is when :

            So, the result is:

          Now substitute back in:

        So, the result is:

      1. Let .

        Then let and substitute :

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. The integral of is when :

          So, the result is:

        Now substitute back in:

      The result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                
 |                                3
 |    5             /        2   \ 
 | cot (x)          \-1 + csc (x)/ 
 | ------- dx = C - ---------------
 |    2                    6       
 | sin (x)                         
 |                                 
/                                  
$$\int \frac{\cot^{5}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\, dx = C - \frac{\left(\csc^{2}{\left(x \right)} - 1\right)^{3}}{6}$$
The graph
The answer [src]
13
--
81
$$\frac{13}{81}$$
=
=
13
--
81
$$\frac{13}{81}$$
13/81
Numerical answer [src]
0.160493827160494
0.160493827160494

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