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Integral of d{x}:
Integral of tan^3xsec^3x
Integral of sin(nx)
Integral of e^x(1-cotx+cosec^2x)dx
Integral of Tan^7(x)
Identical expressions
Tan^ seven (x)
Tan to the power of 7(x)
Tan to the power of seven (x)
Tan7(x)
Tan7x
Tan⁷(x)
Tan^7x
Tan^7(x)dx

Integral of Tan^7(x) dx

The solution

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  1           
  /           
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 |     7      
 |  tan (x) dx
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/             
0

$$\int\limits_{0}^{1} \tan^{7}{\left(x \right)}\, dx$$

Integral(tan(x)^7, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\tan^{7}{\left(x \right)} = \left(\sec^{2}{\left(x \right)} - 1\right)^{3} \tan{\left(x \right)}$
There are multiple ways to do this integral.
Method #1
1. Let $u = \sec^{2}{\left(x \right)}$ .
  
  Then let $du = 2 \tan{\left(x \right)} \sec^{2}{\left(x \right)} dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{u^{3} - 3 u^{2} + 3 u - 1}{4 u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{u^{3} - 3 u^{2} + 3 u - 1}{2 u}\, du = \frac{\int \frac{u^{3} - 3 u^{2} + 3 u - 1}{u}\, du}{2}$
    1. Rewrite the integrand:
      
      $\frac{u^{3} - 3 u^{2} + 3 u - 1}{u} = u^{2} - 3 u + 3 - \frac{1}{u}$
    2. Integrate term-by-term:
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 3 u\right)\, du = - 3 \int u\, du$
      
      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: $- \frac{3 u^{2}}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 3\, du = 3 u$
      
      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: $\frac{u^{3}}{3} - \frac{3 u^{2}}{2} + 3 u - \log{\left(u \right)}$
    So, the result is: $\frac{u^{3}}{6} - \frac{3 u^{2}}{4} + \frac{3 u}{2} - \frac{\log{\left(u \right)}}{2}$
  Now substitute $u$ back in:
  
  $- \frac{\log{\left(\sec^{2}{\left(x \right)} \right)}}{2} + \frac{\sec^{6}{\left(x \right)}}{6} - \frac{3 \sec^{4}{\left(x \right)}}{4} + \frac{3 \sec^{2}{\left(x \right)}}{2}$
Method #2
1. Rewrite the integrand:
  
  $\left(\sec^{2}{\left(x \right)} - 1\right)^{3} \tan{\left(x \right)} = \tan{\left(x \right)} \sec^{6}{\left(x \right)} - 3 \tan{\left(x \right)} \sec^{4}{\left(x \right)} + 3 \tan{\left(x \right)} \sec^{2}{\left(x \right)} - \tan{\left(x \right)}$
2. Integrate term-by-term:
  1. Let $u = \sec^{6}{\left(x \right)}$ .
    
    Then let $du = 6 \tan{\left(x \right)} \sec^{6}{\left(x \right)} dx$ and substitute $\frac{du}{6}$ :
    
    $\int \frac{1}{36}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{6}\, du = \frac{\int 1\, du}{6}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $\frac{u}{6}$
    Now substitute $u$ back in:
    
    $\frac{\sec^{6}{\left(x \right)}}{6}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 3 \tan{\left(x \right)} \sec^{4}{\left(x \right)}\right)\, dx = - 3 \int \tan{\left(x \right)} \sec^{4}{\left(x \right)}\, dx$
    1. Let $u = \sec^{4}{\left(x \right)}$ .
      
      Then let $du = 4 \tan{\left(x \right)} \sec^{4}{\left(x \right)} dx$ and substitute $\frac{du}{4}$ :
      
      $\int \frac{1}{16}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{4}\, du = \frac{\int 1\, du}{4}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $\frac{u}{4}$
      
      Now substitute $u$ back in:
      
      $\frac{\sec^{4}{\left(x \right)}}{4}$
    So, the result is: $- \frac{3 \sec^{4}{\left(x \right)}}{4}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 \tan{\left(x \right)} \sec^{2}{\left(x \right)}\, dx = 3 \int \tan{\left(x \right)} \sec^{2}{\left(x \right)}\, dx$
    1. Let $u = \sec^{2}{\left(x \right)}$ .
      
      Then let $du = 2 \tan{\left(x \right)} \sec^{2}{\left(x \right)} dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{4}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{2}\, du = \frac{\int 1\, du}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $\frac{u}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\sec^{2}{\left(x \right)}}{2}$
    So, the result is: $\frac{3 \sec^{2}{\left(x \right)}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \tan{\left(x \right)}\right)\, dx = - \int \tan{\left(x \right)}\, dx$
    1. Rewrite the integrand:
      
      $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
    2. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int \frac{1}{u}\, du$
      
      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: $- \log{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- \log{\left(\cos{\left(x \right)} \right)}$
    So, the result is: $\log{\left(\cos{\left(x \right)} \right)}$
  The result is: $\log{\left(\cos{\left(x \right)} \right)} + \frac{\sec^{6}{\left(x \right)}}{6} - \frac{3 \sec^{4}{\left(x \right)}}{4} + \frac{3 \sec^{2}{\left(x \right)}}{2}$
Method #3
1. Rewrite the integrand:
  
  $\left(\sec^{2}{\left(x \right)} - 1\right)^{3} \tan{\left(x \right)} = \tan{\left(x \right)} \sec^{6}{\left(x \right)} - 3 \tan{\left(x \right)} \sec^{4}{\left(x \right)} + 3 \tan{\left(x \right)} \sec^{2}{\left(x \right)} - \tan{\left(x \right)}$
2. Integrate term-by-term:
  1. Let $u = \sec^{6}{\left(x \right)}$ .
    
    Then let $du = 6 \tan{\left(x \right)} \sec^{6}{\left(x \right)} dx$ and substitute $\frac{du}{6}$ :
    
    $\int \frac{1}{36}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{6}\, du = \frac{\int 1\, du}{6}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $\frac{u}{6}$
    Now substitute $u$ back in:
    
    $\frac{\sec^{6}{\left(x \right)}}{6}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 3 \tan{\left(x \right)} \sec^{4}{\left(x \right)}\right)\, dx = - 3 \int \tan{\left(x \right)} \sec^{4}{\left(x \right)}\, dx$
    1. Let $u = \sec^{4}{\left(x \right)}$ .
      
      Then let $du = 4 \tan{\left(x \right)} \sec^{4}{\left(x \right)} dx$ and substitute $\frac{du}{4}$ :
      
      $\int \frac{1}{16}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{4}\, du = \frac{\int 1\, du}{4}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $\frac{u}{4}$
      
      Now substitute $u$ back in:
      
      $\frac{\sec^{4}{\left(x \right)}}{4}$
    So, the result is: $- \frac{3 \sec^{4}{\left(x \right)}}{4}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 \tan{\left(x \right)} \sec^{2}{\left(x \right)}\, dx = 3 \int \tan{\left(x \right)} \sec^{2}{\left(x \right)}\, dx$
    1. Let $u = \sec^{2}{\left(x \right)}$ .
      
      Then let $du = 2 \tan{\left(x \right)} \sec^{2}{\left(x \right)} dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{4}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{2}\, du = \frac{\int 1\, du}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $\frac{u}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\sec^{2}{\left(x \right)}}{2}$
    So, the result is: $\frac{3 \sec^{2}{\left(x \right)}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \tan{\left(x \right)}\right)\, dx = - \int \tan{\left(x \right)}\, dx$
    1. Rewrite the integrand:
      
      $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
    2. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int \frac{1}{u}\, du$
      
      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: $- \log{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- \log{\left(\cos{\left(x \right)} \right)}$
    So, the result is: $\log{\left(\cos{\left(x \right)} \right)}$
  The result is: $\log{\left(\cos{\left(x \right)} \right)} + \frac{\sec^{6}{\left(x \right)}}{6} - \frac{3 \sec^{4}{\left(x \right)}}{4} + \frac{3 \sec^{2}{\left(x \right)}}{2}$
Add the constant of integration:

$- \frac{\log{\left(\sec^{2}{\left(x \right)} \right)}}{2} + \frac{\sec^{6}{\left(x \right)}}{6} - \frac{3 \sec^{4}{\left(x \right)}}{4} + \frac{3 \sec^{2}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{\log{\left(\sec^{2}{\left(x \right)} \right)}}{2} + \frac{\sec^{6}{\left(x \right)}}{6} - \frac{3 \sec^{4}{\left(x \right)}}{4} + \frac{3 \sec^{2}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                               
 |                       4         /   2   \      6           2   
 |    7             3*sec (x)   log\sec (x)/   sec (x)   3*sec (x)
 | tan (x) dx = C - --------- - ------------ + ------- + ---------
 |                      4            2            6          2    
/

$${{\log \left(\sin ^2x-1\right)}\over{2}}-{{18\,\sin ^4x-27\,\sin ^2 x+11}\over{12\,\sin ^6x-36\,\sin ^4x+36\,\sin ^2x-12}}$$

Simplify

The answer [src]

                2            4                 
  11   2 - 9*cos (1) + 18*cos (1)              
- -- + -------------------------- + log(cos(1))
  12                 6                         
               12*cos (1)

$${{\log \left(1-\sin ^21\right)}\over{2}}-{{11}\over{12\,\sin ^61-36 \,\sin ^41+36\,\sin ^21-12}}+{{9\,\sin ^21}\over{4\,\sin ^61-12\, \sin ^41+12\,\sin ^21-4}}-{{3\,\sin ^41}\over{2\,\sin ^61-6\,\sin ^4 1+6\,\sin ^21-2}}-{{11}\over{12}}$$

                2            4                 
  11   2 - 9*cos (1) + 18*cos (1)              
- -- + -------------------------- + log(cos(1))
  12                 6                         
               12*cos (1)

$$- \frac{11}{12} + \log{\left(\cos{\left(1 \right)} \right)} + \frac{- 9 \cos^{2}{\left(1 \right)} + 18 \cos^{4}{\left(1 \right)} + 2}{12 \cos^{6}{\left(1 \right)}}$$

Упростить

Numerical answer [src]

1.50462597838128

1.50462597838128

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

Other calculators

How to use it?

Identical expressions

Integral of Tan^7(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3