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Identical expressions
sec^4xtan^4x
sec to the power of 4x tangent of to the power of 4x
sec4xtan4x
sec⁴xtan⁴x
sec^4xtan^4xdx

Integral of sec^4xtan^4x dx

The solution

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  1                   
  /                   
 |                    
 |     4       4      
 |  sec (x)*tan (x) dx
 |                    
/                     
0

$$\int\limits_{0}^{1} \tan^{4}{\left(x \right)} \sec^{4}{\left(x \right)}\, dx$$

Integral(sec(x)^4*tan(x)^4, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\tan^{4}{\left(x \right)} \sec^{4}{\left(x \right)} = \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{4}{\left(x \right)} \sec^{2}{\left(x \right)}$
There are multiple ways to do this integral.
Method #1
1. Let $u = \tan{\left(x \right)}$ .
  
  Then let $du = \left(\tan^{2}{\left(x \right)} + 1\right) dx$ and substitute $du$ :
  
  $\int \left(u^{6} + u^{4}\right)\, du$
  1. Integrate term-by-term:
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
    The result is: $\frac{u^{7}}{7} + \frac{u^{5}}{5}$
  Now substitute $u$ back in:
  
  $\frac{\tan^{7}{\left(x \right)}}{7} + \frac{\tan^{5}{\left(x \right)}}{5}$
Method #2
1. Rewrite the integrand:
  
  $\left(\tan^{2}{\left(x \right)} + 1\right) \tan^{4}{\left(x \right)} \sec^{2}{\left(x \right)} = \tan^{6}{\left(x \right)} \sec^{2}{\left(x \right)} + \tan^{4}{\left(x \right)} \sec^{2}{\left(x \right)}$
2. Integrate term-by-term:
  1. Let $u = \tan{\left(x \right)}$ .
    
    Then let $du = \left(\tan^{2}{\left(x \right)} + 1\right) dx$ and substitute $du$ :
    
    $\int u^{6}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
    Now substitute $u$ back in:
    
    $\frac{\tan^{7}{\left(x \right)}}{7}$
  1. Let $u = \tan{\left(x \right)}$ .
    
    Then let $du = \left(\tan^{2}{\left(x \right)} + 1\right) dx$ and substitute $du$ :
    
    $\int u^{4}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
    Now substitute $u$ back in:
    
    $\frac{\tan^{5}{\left(x \right)}}{5}$
  The result is: $\frac{\tan^{7}{\left(x \right)}}{7} + \frac{\tan^{5}{\left(x \right)}}{5}$
Method #3
1. Rewrite the integrand:
  
  $\left(\tan^{2}{\left(x \right)} + 1\right) \tan^{4}{\left(x \right)} \sec^{2}{\left(x \right)} = \tan^{6}{\left(x \right)} \sec^{2}{\left(x \right)} + \tan^{4}{\left(x \right)} \sec^{2}{\left(x \right)}$
2. Integrate term-by-term:
  1. Let $u = \tan{\left(x \right)}$ .
    
    Then let $du = \left(\tan^{2}{\left(x \right)} + 1\right) dx$ and substitute $du$ :
    
    $\int u^{6}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
    Now substitute $u$ back in:
    
    $\frac{\tan^{7}{\left(x \right)}}{7}$
  1. Let $u = \tan{\left(x \right)}$ .
    
    Then let $du = \left(\tan^{2}{\left(x \right)} + 1\right) dx$ and substitute $du$ :
    
    $\int u^{4}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
    Now substitute $u$ back in:
    
    $\frac{\tan^{5}{\left(x \right)}}{5}$
  The result is: $\frac{\tan^{7}{\left(x \right)}}{7} + \frac{\tan^{5}{\left(x \right)}}{5}$
Add the constant of integration:

$\frac{\tan^{7}{\left(x \right)}}{7} + \frac{\tan^{5}{\left(x \right)}}{5}+ \mathrm{constant}$

The answer is:

$\frac{\tan^{7}{\left(x \right)}}{7} + \frac{\tan^{5}{\left(x \right)}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                          
 |                             5         7   
 |    4       4             tan (x)   tan (x)
 | sec (x)*tan (x) dx = C + ------- + -------
 |                             5         7   
/

$$\int \tan^{4}{\left(x \right)} \sec^{4}{\left(x \right)}\, dx = C + \frac{\tan^{7}{\left(x \right)}}{7} + \frac{\tan^{5}{\left(x \right)}}{5}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   8*sin(1)      sin(1)      sin(1)      2*sin(1)
- ---------- + --------- + ---------- + ---------
        5           7            3      35*cos(1)
  35*cos (1)   7*cos (1)   35*cos (1)

$$- \frac{8 \sin{\left(1 \right)}}{35 \cos^{5}{\left(1 \right)}} + \frac{2 \sin{\left(1 \right)}}{35 \cos{\left(1 \right)}} + \frac{\sin{\left(1 \right)}}{35 \cos^{3}{\left(1 \right)}} + \frac{\sin{\left(1 \right)}}{7 \cos^{7}{\left(1 \right)}}$$

   8*sin(1)      sin(1)      sin(1)      2*sin(1)
- ---------- + --------- + ---------- + ---------
        5           7            3      35*cos(1)
  35*cos (1)   7*cos (1)   35*cos (1)

-8*sin(1)/(35*cos(1)^5) + sin(1)/(7*cos(1)^7) + sin(1)/(35*cos(1)^3) + 2*sin(1)/(35*cos(1))

Numerical answer [src]

5.00730361210706

5.00730361210706

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

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How to use it?

Identical expressions

Integral of sec^4xtan^4x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3