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Derivative of:
cos^23x
Identical expressions
cos^23x
co sinus of e of squared 3x
cos23x
cos²3x
cos to the power of 23x
cos^23xdx
Similar expressions
sinxcos^2(3x)

Integral of cos^23x dx

The solution

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  6            
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 |     23      
 |  cos  (x) dx
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0

$$\int\limits_{0}^{6} \cos^{23}{\left(x \right)}\, dx$$

Integral(cos(x)^23, (x, 0, 6))

Detail solution

Rewrite the integrand:

$\cos^{23}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right)^{11} \cos{\left(x \right)}$
There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(1 - \sin^{2}{\left(x \right)}\right)^{11} \cos{\left(x \right)} = - \sin^{22}{\left(x \right)} \cos{\left(x \right)} + 11 \sin^{20}{\left(x \right)} \cos{\left(x \right)} - 55 \sin^{18}{\left(x \right)} \cos{\left(x \right)} + 165 \sin^{16}{\left(x \right)} \cos{\left(x \right)} - 330 \sin^{14}{\left(x \right)} \cos{\left(x \right)} + 462 \sin^{12}{\left(x \right)} \cos{\left(x \right)} - 462 \sin^{10}{\left(x \right)} \cos{\left(x \right)} + 330 \sin^{8}{\left(x \right)} \cos{\left(x \right)} - 165 \sin^{6}{\left(x \right)} \cos{\left(x \right)} + 55 \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 11 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin^{22}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{22}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{22}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{22}\, du = \frac{u^{23}}{23}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{23}{\left(x \right)}}{23}$
    So, the result is: $- \frac{\sin^{23}{\left(x \right)}}{23}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 11 \sin^{20}{\left(x \right)} \cos{\left(x \right)}\, dx = 11 \int \sin^{20}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{20}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{20}\, du = \frac{u^{21}}{21}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{21}{\left(x \right)}}{21}$
    So, the result is: $\frac{11 \sin^{21}{\left(x \right)}}{21}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 55 \sin^{18}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 55 \int \sin^{18}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{18}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{18}\, du = \frac{u^{19}}{19}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{19}{\left(x \right)}}{19}$
    So, the result is: $- \frac{55 \sin^{19}{\left(x \right)}}{19}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 165 \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx = 165 \int \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{16}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{16}\, du = \frac{u^{17}}{17}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{17}{\left(x \right)}}{17}$
    So, the result is: $\frac{165 \sin^{17}{\left(x \right)}}{17}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 330 \sin^{14}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 330 \int \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{14}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{14}\, du = \frac{u^{15}}{15}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{15}{\left(x \right)}}{15}$
    So, the result is: $- 22 \sin^{15}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 462 \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx = 462 \int \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{12}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{12}\, du = \frac{u^{13}}{13}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{13}{\left(x \right)}}{13}$
    So, the result is: $\frac{462 \sin^{13}{\left(x \right)}}{13}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 462 \sin^{10}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 462 \int \sin^{10}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{10}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{10}\, du = \frac{u^{11}}{11}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{11}{\left(x \right)}}{11}$
    So, the result is: $- 42 \sin^{11}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 330 \sin^{8}{\left(x \right)} \cos{\left(x \right)}\, dx = 330 \int \sin^{8}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{8}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{8}\, du = \frac{u^{9}}{9}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{9}{\left(x \right)}}{9}$
    So, the result is: $\frac{110 \sin^{9}{\left(x \right)}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 165 \sin^{6}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 165 \int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{6}\, du$
      
      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:
      
      $\frac{\sin^{7}{\left(x \right)}}{7}$
    So, the result is: $- \frac{165 \sin^{7}{\left(x \right)}}{7}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 55 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx = 55 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{4}\, du$
      
      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:
      
      $\frac{\sin^{5}{\left(x \right)}}{5}$
    So, the result is: $11 \sin^{5}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 11 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 11 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ 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:
      
      $\frac{\sin^{3}{\left(x \right)}}{3}$
    So, the result is: $- \frac{11 \sin^{3}{\left(x \right)}}{3}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  The result is: $- \frac{\sin^{23}{\left(x \right)}}{23} + \frac{11 \sin^{21}{\left(x \right)}}{21} - \frac{55 \sin^{19}{\left(x \right)}}{19} + \frac{165 \sin^{17}{\left(x \right)}}{17} - 22 \sin^{15}{\left(x \right)} + \frac{462 \sin^{13}{\left(x \right)}}{13} - 42 \sin^{11}{\left(x \right)} + \frac{110 \sin^{9}{\left(x \right)}}{3} - \frac{165 \sin^{7}{\left(x \right)}}{7} + 11 \sin^{5}{\left(x \right)} - \frac{11 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$
Method #2
1. Rewrite the integrand:
  
  $\left(1 - \sin^{2}{\left(x \right)}\right)^{11} \cos{\left(x \right)} = - \sin^{22}{\left(x \right)} \cos{\left(x \right)} + 11 \sin^{20}{\left(x \right)} \cos{\left(x \right)} - 55 \sin^{18}{\left(x \right)} \cos{\left(x \right)} + 165 \sin^{16}{\left(x \right)} \cos{\left(x \right)} - 330 \sin^{14}{\left(x \right)} \cos{\left(x \right)} + 462 \sin^{12}{\left(x \right)} \cos{\left(x \right)} - 462 \sin^{10}{\left(x \right)} \cos{\left(x \right)} + 330 \sin^{8}{\left(x \right)} \cos{\left(x \right)} - 165 \sin^{6}{\left(x \right)} \cos{\left(x \right)} + 55 \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 11 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin^{22}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{22}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{22}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{22}\, du = \frac{u^{23}}{23}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{23}{\left(x \right)}}{23}$
    So, the result is: $- \frac{\sin^{23}{\left(x \right)}}{23}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 11 \sin^{20}{\left(x \right)} \cos{\left(x \right)}\, dx = 11 \int \sin^{20}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{20}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{20}\, du = \frac{u^{21}}{21}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{21}{\left(x \right)}}{21}$
    So, the result is: $\frac{11 \sin^{21}{\left(x \right)}}{21}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 55 \sin^{18}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 55 \int \sin^{18}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{18}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{18}\, du = \frac{u^{19}}{19}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{19}{\left(x \right)}}{19}$
    So, the result is: $- \frac{55 \sin^{19}{\left(x \right)}}{19}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 165 \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx = 165 \int \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{16}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{16}\, du = \frac{u^{17}}{17}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{17}{\left(x \right)}}{17}$
    So, the result is: $\frac{165 \sin^{17}{\left(x \right)}}{17}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 330 \sin^{14}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 330 \int \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{14}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{14}\, du = \frac{u^{15}}{15}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{15}{\left(x \right)}}{15}$
    So, the result is: $- 22 \sin^{15}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 462 \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx = 462 \int \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{12}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{12}\, du = \frac{u^{13}}{13}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{13}{\left(x \right)}}{13}$
    So, the result is: $\frac{462 \sin^{13}{\left(x \right)}}{13}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 462 \sin^{10}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 462 \int \sin^{10}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{10}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{10}\, du = \frac{u^{11}}{11}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{11}{\left(x \right)}}{11}$
    So, the result is: $- 42 \sin^{11}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 330 \sin^{8}{\left(x \right)} \cos{\left(x \right)}\, dx = 330 \int \sin^{8}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{8}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{8}\, du = \frac{u^{9}}{9}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{9}{\left(x \right)}}{9}$
    So, the result is: $\frac{110 \sin^{9}{\left(x \right)}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 165 \sin^{6}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 165 \int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{6}\, du$
      
      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:
      
      $\frac{\sin^{7}{\left(x \right)}}{7}$
    So, the result is: $- \frac{165 \sin^{7}{\left(x \right)}}{7}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 55 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx = 55 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{4}\, du$
      
      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:
      
      $\frac{\sin^{5}{\left(x \right)}}{5}$
    So, the result is: $11 \sin^{5}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 11 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 11 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ 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:
      
      $\frac{\sin^{3}{\left(x \right)}}{3}$
    So, the result is: $- \frac{11 \sin^{3}{\left(x \right)}}{3}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  The result is: $- \frac{\sin^{23}{\left(x \right)}}{23} + \frac{11 \sin^{21}{\left(x \right)}}{21} - \frac{55 \sin^{19}{\left(x \right)}}{19} + \frac{165 \sin^{17}{\left(x \right)}}{17} - 22 \sin^{15}{\left(x \right)} + \frac{462 \sin^{13}{\left(x \right)}}{13} - 42 \sin^{11}{\left(x \right)} + \frac{110 \sin^{9}{\left(x \right)}}{3} - \frac{165 \sin^{7}{\left(x \right)}}{7} + 11 \sin^{5}{\left(x \right)} - \frac{11 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$
Now simplify:

$\frac{\left(- 88179 \sin^{22}{\left(x \right)} + 1062347 \sin^{20}{\left(x \right)} - 5870865 \sin^{18}{\left(x \right)} + 19684665 \sin^{16}{\left(x \right)} - 44618574 \sin^{14}{\left(x \right)} + 72076158 \sin^{12}{\left(x \right)} - 85180914 \sin^{10}{\left(x \right)} + 74364290 \sin^{8}{\left(x \right)} - 47805615 \sin^{6}{\left(x \right)} + 22309287 \sin^{4}{\left(x \right)} - 7436429 \sin^{2}{\left(x \right)} + 2028117\right) \sin{\left(x \right)}}{2028117}$
Add the constant of integration:

$\frac{\left(- 88179 \sin^{22}{\left(x \right)} + 1062347 \sin^{20}{\left(x \right)} - 5870865 \sin^{18}{\left(x \right)} + 19684665 \sin^{16}{\left(x \right)} - 44618574 \sin^{14}{\left(x \right)} + 72076158 \sin^{12}{\left(x \right)} - 85180914 \sin^{10}{\left(x \right)} + 74364290 \sin^{8}{\left(x \right)} - 47805615 \sin^{6}{\left(x \right)} + 22309287 \sin^{4}{\left(x \right)} - 7436429 \sin^{2}{\left(x \right)} + 2028117\right) \sin{\left(x \right)}}{2028117}+ \mathrm{constant}$

The answer is:

$\frac{\left(- 88179 \sin^{22}{\left(x \right)} + 1062347 \sin^{20}{\left(x \right)} - 5870865 \sin^{18}{\left(x \right)} + 19684665 \sin^{16}{\left(x \right)} - 44618574 \sin^{14}{\left(x \right)} + 72076158 \sin^{12}{\left(x \right)} - 85180914 \sin^{10}{\left(x \right)} + 74364290 \sin^{8}{\left(x \right)} - 47805615 \sin^{6}{\left(x \right)} + 22309287 \sin^{4}{\left(x \right)} - 7436429 \sin^{2}{\left(x \right)} + 2028117\right) \sin{\left(x \right)}}{2028117}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                                                                                               
 |                                                                   7            19            3         23            21             9             17             13            
 |    23                   11            15            5      165*sin (x)   55*sin  (x)   11*sin (x)   sin  (x)   11*sin  (x)   110*sin (x)   165*sin  (x)   462*sin  (x)         
 | cos  (x) dx = C - 42*sin  (x) - 22*sin  (x) + 11*sin (x) - ----------- - ----------- - ---------- - -------- + ----------- + ----------- + ------------ + ------------ + sin(x)
 |                                                                 7             19           3           23           21            3             17             13              
/

$$\int \cos^{23}{\left(x \right)}\, dx = C - \frac{\sin^{23}{\left(x \right)}}{23} + \frac{11 \sin^{21}{\left(x \right)}}{21} - \frac{55 \sin^{19}{\left(x \right)}}{19} + \frac{165 \sin^{17}{\left(x \right)}}{17} - 22 \sin^{15}{\left(x \right)} + \frac{462 \sin^{13}{\left(x \right)}}{13} - 42 \sin^{11}{\left(x \right)} + \frac{110 \sin^{9}{\left(x \right)}}{3} - \frac{165 \sin^{7}{\left(x \right)}}{7} + 11 \sin^{5}{\left(x \right)} - \frac{11 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                                                  7            19            3         23            21             9             17             13            
        11            15            5      165*sin (6)   55*sin  (6)   11*sin (6)   sin  (6)   11*sin  (6)   110*sin (6)   165*sin  (6)   462*sin  (6)         
- 42*sin  (6) - 22*sin  (6) + 11*sin (6) - ----------- - ----------- - ---------- - -------- + ----------- + ----------- + ------------ + ------------ + sin(6)
                                                7             19           3           23           21            3             17             13

$$\sin{\left(6 \right)} + 11 \sin^{5}{\left(6 \right)} + \frac{110 \sin^{9}{\left(6 \right)}}{3} + \frac{462 \sin^{13}{\left(6 \right)}}{13} + \frac{165 \sin^{17}{\left(6 \right)}}{17} + \frac{11 \sin^{21}{\left(6 \right)}}{21} - \frac{\sin^{23}{\left(6 \right)}}{23} - \frac{55 \sin^{19}{\left(6 \right)}}{19} - 22 \sin^{15}{\left(6 \right)} - 42 \sin^{11}{\left(6 \right)} - \frac{165 \sin^{7}{\left(6 \right)}}{7} - \frac{11 \sin^{3}{\left(6 \right)}}{3}$$

                                                  7            19            3         23            21             9             17             13            
        11            15            5      165*sin (6)   55*sin  (6)   11*sin (6)   sin  (6)   11*sin  (6)   110*sin (6)   165*sin  (6)   462*sin  (6)         
- 42*sin  (6) - 22*sin  (6) + 11*sin (6) - ----------- - ----------- - ---------- - -------- + ----------- + ----------- + ------------ + ------------ + sin(6)
                                                7             19           3           23           21            3             17             13

-42*sin(6)^11 - 22*sin(6)^15 + 11*sin(6)^5 - 165*sin(6)^7/7 - 55*sin(6)^19/19 - 11*sin(6)^3/3 - sin(6)^23/23 + 11*sin(6)^21/21 + 110*sin(6)^9/3 + 165*sin(6)^17/17 + 462*sin(6)^13/13 + sin(6)

Numerical answer [src]

-0.215376946686499

-0.215376946686499

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

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Identical expressions

Similar expressions

Integral of cos^23x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2