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Integral of d{x}:
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Derivative of:
sin^23x
Identical expressions
sin^23x
sinus of squared 3x
sin23x
sin²3x
sin to the power of 23x
sin^23xdx

Integral of sin^23x dx

The solution

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  2            
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 |     23      
 |  sin  (x) dx
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1

$$\int\limits_{1}^{2} \sin^{23}{\left(x \right)}\, dx$$

Integral(sin(x)^23, (x, 1, 2))

Detail solution

Rewrite the integrand:

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                                                                                                   13             7             17             9            19            3            21         23         23            3            21            19             9             7             17             13            
                11            15            5            5            15            11      462*cos  (2)   165*cos (1)   165*cos  (2)   110*cos (2)   55*cos  (1)   11*cos (1)   11*cos  (2)   cos  (1)   cos  (2)   11*cos (2)   11*cos  (1)   55*cos  (2)   110*cos (1)   165*cos (2)   165*cos  (1)   462*cos  (1)         
-cos(2) - 42*cos  (1) - 22*cos  (1) - 11*cos (2) + 11*cos (1) + 22*cos  (2) + 42*cos  (2) - ------------ - ----------- - ------------ - ----------- - ----------- - ---------- - ----------- - -------- + -------- + ---------- + ----------- + ----------- + ----------- + ----------- + ------------ + ------------ + cos(1)
                                                                                                 13             7             17             3             19           3             21          23         23          3             21            19            3             7             17             13

$$- \frac{11 \cos^{3}{\left(1 \right)}}{3} - \frac{165 \cos^{7}{\left(1 \right)}}{7} + \frac{11 \cos^{3}{\left(2 \right)}}{3} + \frac{165 \cos^{7}{\left(2 \right)}}{7} - 42 \cos^{11}{\left(1 \right)} + 42 \cos^{11}{\left(2 \right)} - 22 \cos^{15}{\left(1 \right)} + 22 \cos^{15}{\left(2 \right)} - \frac{55 \cos^{19}{\left(1 \right)}}{19} + \frac{55 \cos^{19}{\left(2 \right)}}{19} - \frac{\cos^{23}{\left(1 \right)}}{23} + \frac{\cos^{23}{\left(2 \right)}}{23} - \frac{11 \cos^{21}{\left(2 \right)}}{21} + \frac{11 \cos^{21}{\left(1 \right)}}{21} - \frac{165 \cos^{17}{\left(2 \right)}}{17} + \frac{165 \cos^{17}{\left(1 \right)}}{17} - \frac{462 \cos^{13}{\left(2 \right)}}{13} + \frac{462 \cos^{13}{\left(1 \right)}}{13} - \frac{110 \cos^{9}{\left(2 \right)}}{3} - 11 \cos^{5}{\left(2 \right)} + \frac{110 \cos^{9}{\left(1 \right)}}{3} - \cos{\left(2 \right)} + 11 \cos^{5}{\left(1 \right)} + \cos{\left(1 \right)}$$

                                                                                                   13             7             17             9            19            3            21         23         23            3            21            19             9             7             17             13            
                11            15            5            5            15            11      462*cos  (2)   165*cos (1)   165*cos  (2)   110*cos (2)   55*cos  (1)   11*cos (1)   11*cos  (2)   cos  (1)   cos  (2)   11*cos (2)   11*cos  (1)   55*cos  (2)   110*cos (1)   165*cos (2)   165*cos  (1)   462*cos  (1)         
-cos(2) - 42*cos  (1) - 22*cos  (1) - 11*cos (2) + 11*cos (1) + 22*cos  (2) + 42*cos  (2) - ------------ - ----------- - ------------ - ----------- - ----------- - ---------- - ----------- - -------- + -------- + ---------- + ----------- + ----------- + ----------- + ----------- + ------------ + ------------ + cos(1)
                                                                                                 13             7             17             3             19           3             21          23         23          3             21            19            3             7             17             13

-cos(2) - 42*cos(1)^11 - 22*cos(1)^15 - 11*cos(2)^5 + 11*cos(1)^5 + 22*cos(2)^15 + 42*cos(2)^11 - 462*cos(2)^13/13 - 165*cos(1)^7/7 - 165*cos(2)^17/17 - 110*cos(2)^9/3 - 55*cos(1)^19/19 - 11*cos(1)^3/3 - 11*cos(2)^21/21 - cos(1)^23/23 + cos(2)^23/23 + 11*cos(2)^3/3 + 11*cos(1)^21/21 + 55*cos(2)^19/19 + 110*cos(1)^9/3 + 165*cos(2)^7/7 + 165*cos(1)^17/17 + 462*cos(1)^13/13 + cos(1)

Numerical answer [src]

0.506977318641844

0.506977318641844

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

Other calculators

How to use it?

Identical expressions

Integral of sin^23x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2