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Integral of d{x}:
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Integral of 2x^2-1
Integral of tan^3(5x)
Integral of sinx/cos^7x
Identical expressions
tan^ three (5x)
tangent of cubed (5x)
tangent of to the power of three (5x)
tan3(5x)
tan35x
tan³(5x)
tan to the power of 3(5x)
tan^35x
tan^3(5x)dx

Solving integrals/ tan^3(5x)

Integral of tan^3(5x) dx

The solution

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  1             
  /             
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 |     3        
 |  tan (5*x) dx
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0

$$\int\limits_{0}^{1} \tan^{3}{\left(5 x \right)}\, dx$$

Integral(tan(5*x)^3, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\tan^{3}{\left(5 x \right)} = \left(\sec^{2}{\left(5 x \right)} - 1\right) \tan{\left(5 x \right)}$
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 $\frac{du}{10}$ :
  
  $\int \frac{u - 1}{10 u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{u - 1}{u}\, du = \frac{\int \frac{u - 1}{u}\, du}{10}$
    1. Rewrite the integrand:
      
      $\frac{u - 1}{u} = 1 - \frac{1}{u}$
    2. Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{u}\right)\, du = - \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- \log{\left(u \right)}$
      
      The result is: $u - \log{\left(u \right)}$
    So, the result is: $\frac{u}{10} - \frac{\log{\left(u \right)}}{10}$
  Now substitute $u$ back in:
  
  $- \frac{\log{\left(\sec^{2}{\left(5 x \right)} \right)}}{10} + \frac{\sec^{2}{\left(5 x \right)}}{10}$
Method #2
1. Rewrite the integrand:
  
  $\left(\sec^{2}{\left(5 x \right)} - 1\right) \tan{\left(5 x \right)} = \tan{\left(5 x \right)} \sec^{2}{\left(5 x \right)} - \tan{\left(5 x \right)}$
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 $\frac{du}{5}$ :
    
    $\int \frac{u}{5}\, du$
    1. 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:
    
    $\frac{\sec^{2}{\left(5 x \right)}}{10}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \tan{\left(5 x \right)}\right)\, dx = - \int \tan{\left(5 x \right)}\, dx$
    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 $- \frac{du}{5}$ :
      
      $\int \left(- \frac{1}{5 u}\right)\, du$
      
      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:
      
      $- \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}$
    So, the result is: $\frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}$
  The result is: $\frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5} + \frac{\sec^{2}{\left(5 x \right)}}{10}$
Method #3
1. Rewrite the integrand:
  
  $\left(\sec^{2}{\left(5 x \right)} - 1\right) \tan{\left(5 x \right)} = \tan{\left(5 x \right)} \sec^{2}{\left(5 x \right)} - \tan{\left(5 x \right)}$
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 $\frac{du}{5}$ :
    
    $\int \frac{u}{5}\, du$
    1. 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:
    
    $\frac{\sec^{2}{\left(5 x \right)}}{10}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \tan{\left(5 x \right)}\right)\, dx = - \int \tan{\left(5 x \right)}\, dx$
    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 $- \frac{du}{5}$ :
      
      $\int \left(- \frac{1}{5 u}\right)\, du$
      
      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:
      
      $- \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}$
    So, the result is: $\frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}$
  The result is: $\frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5} + \frac{\sec^{2}{\left(5 x \right)}}{10}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                             
 |                       /   2     \      2     
 |    3               log\sec (5*x)/   sec (5*x)
 | tan (5*x) dx = C - -------------- + ---------
 |                          10             10   
/

$$\int \tan^{3}{\left(5 x \right)}\, dx = C - \frac{\log{\left(\sec^{2}{\left(5 x \right)} \right)}}{10} + \frac{\sec^{2}{\left(5 x \right)}}{10}$$

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The graph

Plot the graph f(x)

Numerical answer [src]

26555.9916796074

26555.9916796074

The graph

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

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Identical expressions

Integral of tan^3(5x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3