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Integral of d{x}:
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Identical expressions
(th(x))^ three
(th(x)) cubed
(th(x)) to the power of three
(th(x))3
thx3
(th(x))³
(th(x)) to the power of 3
thx^3
(th(x))^3dx

Solving integrals/ (th(x))^3

Integral of (th(x))^3 dx

The solution

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  1            
  /            
 |             
 |      3      
 |  tanh (x) dx
 |             
/              
0

$$\int\limits_{0}^{1} \tanh^{3}{\left(x \right)}\, dx$$

Integral(tanh(x)^3, (x, 0, 1))

Detail solution

Let $u = \tanh{\left(x \right)}$ .

Then let $du = \left(1 - \tanh^{2}{\left(x \right)}\right) dx$ and substitute $- du$ :

$\int \frac{u^{3}}{u^{2} - 1}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{u^{3}}{u^{2} - 1}\right)\, du = - \int \frac{u^{3}}{u^{2} - 1}\, du$
  1. There are multiple ways to do this integral.
    Method #1
    1. Let $u = u^{2}$ .
      
      Then let $du = 2 u du$ and substitute $du$ :
      
      $\int \frac{u}{2 u - 2}\, du$
      
      Rewrite the integrand:
      
      $\frac{u}{2 u - 2} = \frac{1}{2} + \frac{1}{2 \left(u - 1\right)}$
      
      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:
      
      $\int \frac{1}{2 \left(u - 1\right)}\, du = \frac{\int \frac{1}{u - 1}\, du}{2}$
      
      Let $u = u - 1$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(u - 1 \right)}$
      
      So, the result is: $\frac{\log{\left(u - 1 \right)}}{2}$
      
      The result is: $\frac{u}{2} + \frac{\log{\left(u - 1 \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{u^{2}}{2} + \frac{\log{\left(u^{2} - 1 \right)}}{2}$
    Method #2
    1. Rewrite the integrand:
      
      $\frac{u^{3}}{u^{2} - 1} = u + \frac{1}{2 \left(u + 1\right)} + \frac{1}{2 \left(u - 1\right)}$
    2. Integrate term-by-term:
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{2 \left(u + 1\right)}\, du = \frac{\int \frac{1}{u + 1}\, du}{2}$
      
      Let $u = u + 1$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(u + 1 \right)}$
      
      So, the result is: $\frac{\log{\left(u + 1 \right)}}{2}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{2 \left(u - 1\right)}\, du = \frac{\int \frac{1}{u - 1}\, du}{2}$
      
      Let $u = u - 1$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(u - 1 \right)}$
      
      So, the result is: $\frac{\log{\left(u - 1 \right)}}{2}$
      
      The result is: $\frac{u^{2}}{2} + \frac{\log{\left(u - 1 \right)}}{2} + \frac{\log{\left(u + 1 \right)}}{2}$
  So, the result is: $- \frac{u^{2}}{2} - \frac{\log{\left(u^{2} - 1 \right)}}{2}$
Now substitute $u$ back in:

$- \frac{\log{\left(\tanh^{2}{\left(x \right)} - 1 \right)}}{2} - \frac{\tanh^{2}{\left(x \right)}}{2}$
Now simplify:

$- \frac{\log{\left(- \frac{1}{\cosh^{2}{\left(x \right)}} \right)}}{2} - \frac{\tanh^{2}{\left(x \right)}}{2}$
Add the constant of integration:

$- \frac{\log{\left(- \frac{1}{\cosh^{2}{\left(x \right)}} \right)}}{2} - \frac{\tanh^{2}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{\log{\left(- \frac{1}{\cosh^{2}{\left(x \right)}} \right)}}{2} - \frac{\tanh^{2}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                               
 |                       2         /         2   \
 |     3             tanh (x)   log\-1 + tanh (x)/
 | tanh (x) dx = C - -------- - ------------------
 |                      2               2         
/

$$\int \tanh^{3}{\left(x \right)}\, dx = C - \frac{\log{\left(\tanh^{2}{\left(x \right)} - 1 \right)}}{2} - \frac{\tanh^{2}{\left(x \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                           2   
                       tanh (1)
1 - log(1 + tanh(1)) - --------
                          2

$$- \log{\left(\tanh{\left(1 \right)} + 1 \right)} - \frac{\tanh^{2}{\left(1 \right)}}{2} + 1$$

                           2   
                       tanh (1)
1 - log(1 + tanh(1)) - --------
                          2

$$- \log{\left(\tanh{\left(1 \right)} + 1 \right)} - \frac{\tanh^{2}{\left(1 \right)}}{2} + 1$$

Упростить

Numerical answer [src]

0.14376800129004

0.14376800129004

The graph

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

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How to use it?

Identical expressions

Integral of (th(x))^3 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2