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tg^3(x/3)

Integral of tg^3(x/3) dx

Limits of integration:

from to
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The graph:

from to

Piecewise:

The solution

You have entered [src]
  1           
  /           
 |            
 |     3/x\   
 |  tan |-| dx
 |      \3/   
 |            
/             
0             
$$\int\limits_{0}^{1} \tan^{3}{\left(\frac{x}{3} \right)}\, dx$$
Integral(tan(x/3)^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. Let .

          Then let and substitute :

          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:

          Now substitute back in:

        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 a constant is the constant times the variable of integration:

          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 a constant times a function is the constant times the integral of the function:

            1. The integral of is .

            So, the result 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 a constant is the constant times the variable of integration:

          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 a constant times a function is the constant times the integral of the function:

            1. The integral of is .

            So, the result 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/x\\        2/x\
 |                  3*log|3*sec |-||   3*sec |-|
 |    3/x\               \      \3//         \3/
 | tan |-| dx = C - ---------------- + ---------
 |     \3/                 2               2    
 |                                              
/                                               
$$3\,\left({{\log \left(\sin ^2\left({{x}\over{3}}\right)-1\right) }\over{2}}-{{1}\over{2\,\sin ^2\left({{x}\over{3}}\right)-2}}\right)$$
The graph
The answer [src]
  3                          3     
- - + 3*log(cos(1/3)) + -----------
  2                          2     
                        2*cos (1/3)
$$3\,\left({{\log \left(1-\sin ^2\left({{1}\over{3}}\right)\right) }\over{2}}-{{1}\over{2\,\sin ^2\left({{1}\over{3}}\right)-2}}-{{1 }\over{2}}\right)$$
=
=
  3                          3     
- - + 3*log(cos(1/3)) + -----------
  2                          2     
                        2*cos (1/3)
$$- \frac{3}{2} + 3 \log{\left(\cos{\left(\frac{1}{3} \right)} \right)} + \frac{3}{2 \cos^{2}{\left(\frac{1}{3} \right)}}$$
Numerical answer [src]
0.00998954487832324
0.00998954487832324
The graph
Integral of tg^3(x/3) dx

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