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Identical expressions
tan(five x)^5
tangent of (5x) to the power of 5
tangent of (five x) to the power of 5
tan(5x)5
tan5x5
tan(5x)⁵
tan5x^5
tan(5x)^5dx

Integral of tan(5x)^5 dx

The solution

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  1             
  /             
 |              
 |     5        
 |  tan (5*x) dx
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0

$$\int\limits_{0}^{1} \tan^{5}{\left(5 x \right)}\, dx$$

Integral(tan(5*x)^5, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\tan^{5}{\left(5 x \right)} = \left(\sec^{2}{\left(5 x \right)} - 1\right)^{2} \tan{\left(5 x \right)}$
There are multiple ways to do this integral.
Method #1
1. Let $u = \sec^{2}{\left(5 x \right)}$ .
  
  Then let $du = 10 \tan{\left(5 x \right)} \sec^{2}{\left(5 x \right)} dx$ and substitute $\frac{du}{10}$ :
  
  $\int \frac{u^{2} - 2 u + 1}{10 u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{u^{2} - 2 u + 1}{u}\, du = \frac{\int \frac{u^{2} - 2 u + 1}{u}\, du}{10}$
    1. Rewrite the integrand:
      
      $\frac{u^{2} - 2 u + 1}{u} = u - 2 + \frac{1}{u}$
    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 is the constant times the variable of integration:
      
      $\int \left(-2\right)\, du = - 2 u$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      The result is: $\frac{u^{2}}{2} - 2 u + \log{\left(u \right)}$
    So, the result is: $\frac{u^{2}}{20} - \frac{u}{5} + \frac{\log{\left(u \right)}}{10}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(\sec^{2}{\left(5 x \right)} \right)}}{10} + \frac{\sec^{4}{\left(5 x \right)}}{20} - \frac{\sec^{2}{\left(5 x \right)}}{5}$
Method #2
1. Rewrite the integrand:
  
  $\left(\sec^{2}{\left(5 x \right)} - 1\right)^{2} \tan{\left(5 x \right)} = \tan{\left(5 x \right)} \sec^{4}{\left(5 x \right)} - 2 \tan{\left(5 x \right)} \sec^{2}{\left(5 x \right)} + \tan{\left(5 x \right)}$
2. Integrate term-by-term:
  1. Let $u = \sec{\left(5 x \right)}$ .
    
    Then let $du = 5 \tan{\left(5 x \right)} \sec{\left(5 x \right)} dx$ and substitute $\frac{du}{5}$ :
    
    $\int \frac{u^{3}}{5}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{3}\, du = \frac{\int u^{3}\, du}{5}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{3}\, du = \frac{u^{4}}{4}$
      
      So, the result is: $\frac{u^{4}}{20}$
    Now substitute $u$ back in:
    
    $\frac{\sec^{4}{\left(5 x \right)}}{20}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 \tan{\left(5 x \right)} \sec^{2}{\left(5 x \right)}\right)\, dx = - 2 \int \tan{\left(5 x \right)} \sec^{2}{\left(5 x \right)}\, dx$
    1. Let $u = \sec{\left(5 x \right)}$ .
      
      Then let $du = 5 \tan{\left(5 x \right)} \sec{\left(5 x \right)} dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{u}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = \frac{\int u\, du}{5}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $\frac{u^{2}}{10}$
      
      Now substitute $u$ back in:
      
      $\frac{\sec^{2}{\left(5 x \right)}}{10}$
    So, the result is: $- \frac{\sec^{2}{\left(5 x \right)}}{5}$
  1. Rewrite the integrand:
    
    $\tan{\left(5 x \right)} = \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$
  2. Let $u = \cos{\left(5 x \right)}$ .
    
    Then let $du = - 5 \sin{\left(5 x \right)} dx$ and substitute $- \frac{du}{5}$ :
    
    $\int \left(- \frac{1}{5 u}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = - \frac{\int \frac{1}{u}\, du}{5}$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- \frac{\log{\left(u \right)}}{5}$
    Now substitute $u$ back in:
    
    $- \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}$
  The result is: $- \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5} + \frac{\sec^{4}{\left(5 x \right)}}{20} - \frac{\sec^{2}{\left(5 x \right)}}{5}$
Method #3
1. Rewrite the integrand:
  
  $\left(\sec^{2}{\left(5 x \right)} - 1\right)^{2} \tan{\left(5 x \right)} = \tan{\left(5 x \right)} \sec^{4}{\left(5 x \right)} - 2 \tan{\left(5 x \right)} \sec^{2}{\left(5 x \right)} + \tan{\left(5 x \right)}$
2. Integrate term-by-term:
  1. Let $u = \sec{\left(5 x \right)}$ .
    
    Then let $du = 5 \tan{\left(5 x \right)} \sec{\left(5 x \right)} dx$ and substitute $\frac{du}{5}$ :
    
    $\int \frac{u^{3}}{5}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{3}\, du = \frac{\int u^{3}\, du}{5}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{3}\, du = \frac{u^{4}}{4}$
      
      So, the result is: $\frac{u^{4}}{20}$
    Now substitute $u$ back in:
    
    $\frac{\sec^{4}{\left(5 x \right)}}{20}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 \tan{\left(5 x \right)} \sec^{2}{\left(5 x \right)}\right)\, dx = - 2 \int \tan{\left(5 x \right)} \sec^{2}{\left(5 x \right)}\, dx$
    1. Let $u = \sec{\left(5 x \right)}$ .
      
      Then let $du = 5 \tan{\left(5 x \right)} \sec{\left(5 x \right)} dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{u}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = \frac{\int u\, du}{5}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $\frac{u^{2}}{10}$
      
      Now substitute $u$ back in:
      
      $\frac{\sec^{2}{\left(5 x \right)}}{10}$
    So, the result is: $- \frac{\sec^{2}{\left(5 x \right)}}{5}$
  1. Rewrite the integrand:
    
    $\tan{\left(5 x \right)} = \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$
  2. Let $u = \cos{\left(5 x \right)}$ .
    
    Then let $du = - 5 \sin{\left(5 x \right)} dx$ and substitute $- \frac{du}{5}$ :
    
    $\int \left(- \frac{1}{5 u}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = - \frac{\int \frac{1}{u}\, du}{5}$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- \frac{\log{\left(u \right)}}{5}$
    Now substitute $u$ back in:
    
    $- \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}$
  The result is: $- \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5} + \frac{\sec^{4}{\left(5 x \right)}}{20} - \frac{\sec^{2}{\left(5 x \right)}}{5}$
Add the constant of integration:

$\frac{\log{\left(\sec^{2}{\left(5 x \right)} \right)}}{10} + \frac{\sec^{4}{\left(5 x \right)}}{20} - \frac{\sec^{2}{\left(5 x \right)}}{5}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(\sec^{2}{\left(5 x \right)} \right)}}{10} + \frac{\sec^{4}{\left(5 x \right)}}{20} - \frac{\sec^{2}{\left(5 x \right)}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                         
 |                       2           /   2     \      4     
 |    5               sec (5*x)   log\sec (5*x)/   sec (5*x)
 | tan (5*x) dx = C - --------- + -------------- + ---------
 |                        5             10             20   
/

$$\int \tan^{5}{\left(5 x \right)}\, dx = C + \frac{\log{\left(\sec^{2}{\left(5 x \right)} \right)}}{10} + \frac{\sec^{4}{\left(5 x \right)}}{20} - \frac{\sec^{2}{\left(5 x \right)}}{5}$$

Simplify

The graph

Plot the graph f(x)

Numerical answer [src]

872475987.400727

872475987.400727

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

Other calculators

How to use it?

Identical expressions

Integral of tan(5x)^5 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3