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Integral of d{x}:
Integral of x^2*dx
Integral of x/(x-1)
Integral of x/(x-1)^2
Integral of 10x
Identical expressions
sec(x)tan^ five (x)
sec(x) tangent of to the power of 5(x)
sec(x) tangent of to the power of five (x)
sec(x)tan5(x)
secxtan5x
sec(x)tan⁵(x)
secxtan^5x
sec(x)tan^5(x)dx

Integral of sec(x)tan^5(x) dx

The solution

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

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

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

Detail solution

Rewrite the integrand:

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

$\frac{\sec^{5}{\left(x \right)}}{5} - \frac{2 \sec^{3}{\left(x \right)}}{3} + \sec{\left(x \right)}+ \mathrm{constant}$

The answer is:

$\frac{\sec^{5}{\left(x \right)}}{5} - \frac{2 \sec^{3}{\left(x \right)}}{3} + \sec{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                 2            4   
  8    3 - 10*cos (1) + 15*cos (1)
- -- + ---------------------------
  15                  5           
                15*cos (1)

$$- \frac{8}{15} + \frac{- 10 \cos^{2}{\left(1 \right)} + 15 \cos^{4}{\left(1 \right)} + 3}{15 \cos^{5}{\left(1 \right)}}$$

                 2            4   
  8    3 - 10*cos (1) + 15*cos (1)
- -- + ---------------------------
  15                  5           
                15*cos (1)

$$- \frac{8}{15} + \frac{- 10 \cos^{2}{\left(1 \right)} + 15 \cos^{4}{\left(1 \right)} + 3}{15 \cos^{5}{\left(1 \right)}}$$

-8/15 + (3 - 10*cos(1)^2 + 15*cos(1)^4)/(15*cos(1)^5)

Numerical answer [src]

1.43437365760108

1.43437365760108

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

Other calculators

How to use it?

Identical expressions

Integral of sec(x)tan^5(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3