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Integral of cot^3(x) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src]
  1           
  /           
 |            
 |     3      
 |  cot (x) dx
 |            
/             
0             
$$\int\limits_{0}^{1} \cot^{3}{\left(x \right)}\, dx$$
Integral(cot(x)^3, (x, 0, 1))
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. Rewrite the integrand:

        2. Integrate term-by-term:

          1. The integral of a constant is the constant times the variable of integration:

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

            1. The integral of is .

            So, the result is:

          The result is:

        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. Rewrite the integrand:

        2. Let .

          Then let and substitute :

          1. The integral of is .

          Now substitute back in:

        So, the result is:

      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. Rewrite the integrand:

        2. Let .

          Then let and substitute :

          1. The integral of is .

          Now substitute back in:

        So, the result is:

      The result is:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                       
 |                     /   2   \      2   
 |    3             log\csc (x)/   csc (x)
 | cot (x) dx = C + ------------ - -------
 |                       2            2   
/                                         
$$\int \cot^{3}{\left(x \right)}\, dx = C + \frac{\log{\left(\csc^{2}{\left(x \right)} \right)}}{2} - \frac{\csc^{2}{\left(x \right)}}{2}$$
The answer [src]
oo
$$\infty$$
=
=
oo
$$\infty$$
oo
Numerical answer [src]
9.15365037903492e+37
9.15365037903492e+37

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