Integral of sin²5x dx

The solution

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0

$$\int\limits_{0}^{1} \sin^{25}{\left(x \right)}\, dx$$

Integral(sin(x)^25, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\sin^{25}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right)^{12} \sin{\left(x \right)}$
There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(1 - \cos^{2}{\left(x \right)}\right)^{12} \sin{\left(x \right)} = \sin{\left(x \right)} \cos^{24}{\left(x \right)} - 12 \sin{\left(x \right)} \cos^{22}{\left(x \right)} + 66 \sin{\left(x \right)} \cos^{20}{\left(x \right)} - 220 \sin{\left(x \right)} \cos^{18}{\left(x \right)} + 495 \sin{\left(x \right)} \cos^{16}{\left(x \right)} - 792 \sin{\left(x \right)} \cos^{14}{\left(x \right)} + 924 \sin{\left(x \right)} \cos^{12}{\left(x \right)} - 792 \sin{\left(x \right)} \cos^{10}{\left(x \right)} + 495 \sin{\left(x \right)} \cos^{8}{\left(x \right)} - 220 \sin{\left(x \right)} \cos^{6}{\left(x \right)} + 66 \sin{\left(x \right)} \cos^{4}{\left(x \right)} - 12 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \sin{\left(x \right)}$
2. Integrate term-by-term:
  1. Let $u = \cos{\left(x \right)}$ .
    
    Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
    
    $\int \left(- u^{24}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{24}\, du = - \int u^{24}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{24}\, du = \frac{u^{25}}{25}$
      
      So, the result is: $- \frac{u^{25}}{25}$
    Now substitute $u$ back in:
    
    $- \frac{\cos^{25}{\left(x \right)}}{25}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 12 \sin{\left(x \right)} \cos^{22}{\left(x \right)}\right)\, dx = - 12 \int \sin{\left(x \right)} \cos^{22}{\left(x \right)}\, dx$
    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: $\frac{12 \cos^{23}{\left(x \right)}}{23}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 66 \sin{\left(x \right)} \cos^{20}{\left(x \right)}\, dx = 66 \int \sin{\left(x \right)} \cos^{20}{\left(x \right)}\, dx$
    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: $- \frac{22 \cos^{21}{\left(x \right)}}{7}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 220 \sin{\left(x \right)} \cos^{18}{\left(x \right)}\right)\, dx = - 220 \int \sin{\left(x \right)} \cos^{18}{\left(x \right)}\, dx$
    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: $\frac{220 \cos^{19}{\left(x \right)}}{19}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 495 \sin{\left(x \right)} \cos^{16}{\left(x \right)}\, dx = 495 \int \sin{\left(x \right)} \cos^{16}{\left(x \right)}\, dx$
    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: $- \frac{495 \cos^{17}{\left(x \right)}}{17}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 792 \sin{\left(x \right)} \cos^{14}{\left(x \right)}\right)\, dx = - 792 \int \sin{\left(x \right)} \cos^{14}{\left(x \right)}\, dx$
    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: $\frac{264 \cos^{15}{\left(x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 924 \sin{\left(x \right)} \cos^{12}{\left(x \right)}\, dx = 924 \int \sin{\left(x \right)} \cos^{12}{\left(x \right)}\, dx$
    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: $- \frac{924 \cos^{13}{\left(x \right)}}{13}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 792 \sin{\left(x \right)} \cos^{10}{\left(x \right)}\right)\, dx = - 792 \int \sin{\left(x \right)} \cos^{10}{\left(x \right)}\, dx$
    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: $72 \cos^{11}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 495 \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx = 495 \int \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx$
    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: $- 55 \cos^{9}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 220 \sin{\left(x \right)} \cos^{6}{\left(x \right)}\right)\, dx = - 220 \int \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx$
    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: $\frac{220 \cos^{7}{\left(x \right)}}{7}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 66 \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx = 66 \int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx$
    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: $- \frac{66 \cos^{5}{\left(x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 12 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 12 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$
    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: $4 \cos^{3}{\left(x \right)}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
  The result is: $- \frac{\cos^{25}{\left(x \right)}}{25} + \frac{12 \cos^{23}{\left(x \right)}}{23} - \frac{22 \cos^{21}{\left(x \right)}}{7} + \frac{220 \cos^{19}{\left(x \right)}}{19} - \frac{495 \cos^{17}{\left(x \right)}}{17} + \frac{264 \cos^{15}{\left(x \right)}}{5} - \frac{924 \cos^{13}{\left(x \right)}}{13} + 72 \cos^{11}{\left(x \right)} - 55 \cos^{9}{\left(x \right)} + \frac{220 \cos^{7}{\left(x \right)}}{7} - \frac{66 \cos^{5}{\left(x \right)}}{5} + 4 \cos^{3}{\left(x \right)} - \cos{\left(x \right)}$
Method #2
1. Rewrite the integrand:
  
  $\left(1 - \cos^{2}{\left(x \right)}\right)^{12} \sin{\left(x \right)} = \sin{\left(x \right)} \cos^{24}{\left(x \right)} - 12 \sin{\left(x \right)} \cos^{22}{\left(x \right)} + 66 \sin{\left(x \right)} \cos^{20}{\left(x \right)} - 220 \sin{\left(x \right)} \cos^{18}{\left(x \right)} + 495 \sin{\left(x \right)} \cos^{16}{\left(x \right)} - 792 \sin{\left(x \right)} \cos^{14}{\left(x \right)} + 924 \sin{\left(x \right)} \cos^{12}{\left(x \right)} - 792 \sin{\left(x \right)} \cos^{10}{\left(x \right)} + 495 \sin{\left(x \right)} \cos^{8}{\left(x \right)} - 220 \sin{\left(x \right)} \cos^{6}{\left(x \right)} + 66 \sin{\left(x \right)} \cos^{4}{\left(x \right)} - 12 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \sin{\left(x \right)}$
2. Integrate term-by-term:
  1. Let $u = \cos{\left(x \right)}$ .
    
    Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
    
    $\int \left(- u^{24}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{24}\, du = - \int u^{24}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{24}\, du = \frac{u^{25}}{25}$
      
      So, the result is: $- \frac{u^{25}}{25}$
    Now substitute $u$ back in:
    
    $- \frac{\cos^{25}{\left(x \right)}}{25}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 12 \sin{\left(x \right)} \cos^{22}{\left(x \right)}\right)\, dx = - 12 \int \sin{\left(x \right)} \cos^{22}{\left(x \right)}\, dx$
    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: $\frac{12 \cos^{23}{\left(x \right)}}{23}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 66 \sin{\left(x \right)} \cos^{20}{\left(x \right)}\, dx = 66 \int \sin{\left(x \right)} \cos^{20}{\left(x \right)}\, dx$
    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: $- \frac{22 \cos^{21}{\left(x \right)}}{7}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 220 \sin{\left(x \right)} \cos^{18}{\left(x \right)}\right)\, dx = - 220 \int \sin{\left(x \right)} \cos^{18}{\left(x \right)}\, dx$
    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: $\frac{220 \cos^{19}{\left(x \right)}}{19}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 495 \sin{\left(x \right)} \cos^{16}{\left(x \right)}\, dx = 495 \int \sin{\left(x \right)} \cos^{16}{\left(x \right)}\, dx$
    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: $- \frac{495 \cos^{17}{\left(x \right)}}{17}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 792 \sin{\left(x \right)} \cos^{14}{\left(x \right)}\right)\, dx = - 792 \int \sin{\left(x \right)} \cos^{14}{\left(x \right)}\, dx$
    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: $\frac{264 \cos^{15}{\left(x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 924 \sin{\left(x \right)} \cos^{12}{\left(x \right)}\, dx = 924 \int \sin{\left(x \right)} \cos^{12}{\left(x \right)}\, dx$
    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: $- \frac{924 \cos^{13}{\left(x \right)}}{13}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 792 \sin{\left(x \right)} \cos^{10}{\left(x \right)}\right)\, dx = - 792 \int \sin{\left(x \right)} \cos^{10}{\left(x \right)}\, dx$
    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: $72 \cos^{11}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 495 \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx = 495 \int \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx$
    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: $- 55 \cos^{9}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 220 \sin{\left(x \right)} \cos^{6}{\left(x \right)}\right)\, dx = - 220 \int \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx$
    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: $\frac{220 \cos^{7}{\left(x \right)}}{7}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 66 \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx = 66 \int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx$
    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: $- \frac{66 \cos^{5}{\left(x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 12 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 12 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$
    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: $4 \cos^{3}{\left(x \right)}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
  The result is: $- \frac{\cos^{25}{\left(x \right)}}{25} + \frac{12 \cos^{23}{\left(x \right)}}{23} - \frac{22 \cos^{21}{\left(x \right)}}{7} + \frac{220 \cos^{19}{\left(x \right)}}{19} - \frac{495 \cos^{17}{\left(x \right)}}{17} + \frac{264 \cos^{15}{\left(x \right)}}{5} - \frac{924 \cos^{13}{\left(x \right)}}{13} + 72 \cos^{11}{\left(x \right)} - 55 \cos^{9}{\left(x \right)} + \frac{220 \cos^{7}{\left(x \right)}}{7} - \frac{66 \cos^{5}{\left(x \right)}}{5} + 4 \cos^{3}{\left(x \right)} - \cos{\left(x \right)}$
Now simplify:

$\frac{\left(- 676039 \cos^{24}{\left(x \right)} + 8817900 \cos^{22}{\left(x \right)} - 53117350 \cos^{20}{\left(x \right)} + 195695500 \cos^{18}{\left(x \right)} - 492116625 \cos^{16}{\left(x \right)} + 892371480 \cos^{14}{\left(x \right)} - 1201269300 \cos^{12}{\left(x \right)} + 1216870200 \cos^{10}{\left(x \right)} - 929553625 \cos^{8}{\left(x \right)} + 531173500 \cos^{6}{\left(x \right)} - 223092870 \cos^{4}{\left(x \right)} + 67603900 \cos^{2}{\left(x \right)} - 16900975\right) \cos{\left(x \right)}}{16900975}$
Add the constant of integration:

$\frac{\left(- 676039 \cos^{24}{\left(x \right)} + 8817900 \cos^{22}{\left(x \right)} - 53117350 \cos^{20}{\left(x \right)} + 195695500 \cos^{18}{\left(x \right)} - 492116625 \cos^{16}{\left(x \right)} + 892371480 \cos^{14}{\left(x \right)} - 1201269300 \cos^{12}{\left(x \right)} + 1216870200 \cos^{10}{\left(x \right)} - 929553625 \cos^{8}{\left(x \right)} + 531173500 \cos^{6}{\left(x \right)} - 223092870 \cos^{4}{\left(x \right)} + 67603900 \cos^{2}{\left(x \right)} - 16900975\right) \cos{\left(x \right)}}{16900975}+ \mathrm{constant}$

The answer is:

$\frac{\left(- 676039 \cos^{24}{\left(x \right)} + 8817900 \cos^{22}{\left(x \right)} - 53117350 \cos^{20}{\left(x \right)} + 195695500 \cos^{18}{\left(x \right)} - 492116625 \cos^{16}{\left(x \right)} + 892371480 \cos^{14}{\left(x \right)} - 1201269300 \cos^{12}{\left(x \right)} + 1216870200 \cos^{10}{\left(x \right)} - 929553625 \cos^{8}{\left(x \right)} + 531173500 \cos^{6}{\left(x \right)} - 223092870 \cos^{4}{\left(x \right)} + 67603900 \cos^{2}{\left(x \right)} - 16900975\right) \cos{\left(x \right)}}{16900975}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                                                                                                             
 |                                                                          13             17            5            21         25            23             7             19             15   
 |    25                            9           3            11      924*cos  (x)   495*cos  (x)   66*cos (x)   22*cos  (x)   cos  (x)   12*cos  (x)   220*cos (x)   220*cos  (x)   264*cos  (x)
 | sin  (x) dx = C - cos(x) - 55*cos (x) + 4*cos (x) + 72*cos  (x) - ------------ - ------------ - ---------- - ----------- - -------- + ----------- + ----------- + ------------ + ------------
 |                                                                        13             17            5             7           25           23            7             19             5      
/

$$\int \sin^{25}{\left(x \right)}\, dx = C - \frac{\cos^{25}{\left(x \right)}}{25} + \frac{12 \cos^{23}{\left(x \right)}}{23} - \frac{22 \cos^{21}{\left(x \right)}}{7} + \frac{220 \cos^{19}{\left(x \right)}}{19} - \frac{495 \cos^{17}{\left(x \right)}}{17} + \frac{264 \cos^{15}{\left(x \right)}}{5} - \frac{924 \cos^{13}{\left(x \right)}}{13} + 72 \cos^{11}{\left(x \right)} - 55 \cos^{9}{\left(x \right)} + \frac{220 \cos^{7}{\left(x \right)}}{7} - \frac{66 \cos^{5}{\left(x \right)}}{5} + 4 \cos^{3}{\left(x \right)} - \cos{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                                                                  13             17            5            21         25            23             7             19             15   
4194304                   9           3            11      924*cos  (1)   495*cos  (1)   66*cos (1)   22*cos  (1)   cos  (1)   12*cos  (1)   220*cos (1)   220*cos  (1)   264*cos  (1)
-------- - cos(1) - 55*cos (1) + 4*cos (1) + 72*cos  (1) - ------------ - ------------ - ---------- - ----------- - -------- + ----------- + ----------- + ------------ + ------------
16900975                                                        13             17            5             7           25           23            7             19             5

$$- \frac{66 \cos^{5}{\left(1 \right)}}{5} - \cos{\left(1 \right)} - 55 \cos^{9}{\left(1 \right)} - \frac{924 \cos^{13}{\left(1 \right)}}{13} - \frac{495 \cos^{17}{\left(1 \right)}}{17} - \frac{22 \cos^{21}{\left(1 \right)}}{7} - \frac{\cos^{25}{\left(1 \right)}}{25} + \frac{12 \cos^{23}{\left(1 \right)}}{23} + \frac{220 \cos^{19}{\left(1 \right)}}{19} + \frac{264 \cos^{15}{\left(1 \right)}}{5} + 72 \cos^{11}{\left(1 \right)} + \frac{4194304}{16900975} + \frac{220 \cos^{7}{\left(1 \right)}}{7} + 4 \cos^{3}{\left(1 \right)}$$

                                                                  13             17            5            21         25            23             7             19             15   
4194304                   9           3            11      924*cos  (1)   495*cos  (1)   66*cos (1)   22*cos  (1)   cos  (1)   12*cos  (1)   220*cos (1)   220*cos  (1)   264*cos  (1)
-------- - cos(1) - 55*cos (1) + 4*cos (1) + 72*cos  (1) - ------------ - ------------ - ---------- - ----------- - -------- + ----------- + ----------- + ------------ + ------------
16900975                                                        13             17            5             7           25           23            7             19             5

4194304/16900975 - cos(1) - 55*cos(1)^9 + 4*cos(1)^3 + 72*cos(1)^11 - 924*cos(1)^13/13 - 495*cos(1)^17/17 - 66*cos(1)^5/5 - 22*cos(1)^21/7 - cos(1)^25/25 + 12*cos(1)^23/23 + 220*cos(1)^7/7 + 220*cos(1)^19/19 + 264*cos(1)^15/5

Numerical answer [src]

0.000743706011186276

0.000743706011186276

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of sin²5x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2