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Integral of d{x}:
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Identical expressions
(8x⁵- three ÷sin²4x+ two ÷3x²)dx
(8x⁵ minus 3÷ sinus of ²4x plus 2÷3x²)dx
(8x⁵ minus three ÷ sinus of ²4x plus two ÷3x²)dx
8x⁵-3÷sin²4x+2÷3x²dx
Similar expressions
(8x⁵-3÷sin²4x-2÷3x²)dx
(8x⁵+3÷sin²4x+2÷3x²)dx

Solving integrals/ (8x⁵-3÷sin²4x+2÷3x²)dx

Integral of (8x⁵-3÷sin²4x+2÷3x²)dx dx

The solution

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 |  |   5      3       2*x |   
 |  |8*x  - -------- + ----| dx
 |  |          24       3  |   
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0

$$\int\limits_{0}^{1} \left(\frac{2 x^{2}}{3} + \left(8 x^{5} - \frac{3}{\sin^{24}{\left(x \right)}}\right)\right)\, dx$$

Integral(8*x^5 - 3/sin(x)^24 + 2*x^2/3, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 x^{2}}{3}\, dx = \frac{2 \int x^{2}\, dx}{3}$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  So, the result is: $\frac{2 x^{3}}{9}$
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 8 x^{5}\, dx = 8 \int x^{5}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{5}\, dx = \frac{x^{6}}{6}$
    So, the result is: $\frac{4 x^{6}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{3}{\sin^{24}{\left(x \right)}}\right)\, dx = - 3 \int \frac{1}{\sin^{24}{\left(x \right)}}\, dx$
    1. Rewrite the integrand:
      
      $\csc^{24}{\left(x \right)} = \left(\cot^{2}{\left(x \right)} + 1\right)^{11} \csc^{2}{\left(x \right)}$
    2. There are multiple ways to do this integral.
      Method #1
      
      Rewrite the integrand:
      
      $\left(\cot^{2}{\left(x \right)} + 1\right)^{11} \csc^{2}{\left(x \right)} = \cot^{22}{\left(x \right)} \csc^{2}{\left(x \right)} + 11 \cot^{20}{\left(x \right)} \csc^{2}{\left(x \right)} + 55 \cot^{18}{\left(x \right)} \csc^{2}{\left(x \right)} + 165 \cot^{16}{\left(x \right)} \csc^{2}{\left(x \right)} + 330 \cot^{14}{\left(x \right)} \csc^{2}{\left(x \right)} + 462 \cot^{12}{\left(x \right)} \csc^{2}{\left(x \right)} + 462 \cot^{10}{\left(x \right)} \csc^{2}{\left(x \right)} + 330 \cot^{8}{\left(x \right)} \csc^{2}{\left(x \right)} + 165 \cot^{6}{\left(x \right)} \csc^{2}{\left(x \right)} + 55 \cot^{4}{\left(x \right)} \csc^{2}{\left(x \right)} + 11 \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)} + \csc^{2}{\left(x \right)}$
      
      Integrate term-by-term:
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{23}{\left(x \right)}}{23}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 11 \cot^{20}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 11 \int \cot^{20}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{21}{\left(x \right)}}{21}$
      
      So, the result is: $- \frac{11 \cot^{21}{\left(x \right)}}{21}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 55 \cot^{18}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 55 \int \cot^{18}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{19}{\left(x \right)}}{19}$
      
      So, the result is: $- \frac{55 \cot^{19}{\left(x \right)}}{19}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 165 \cot^{16}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 165 \int \cot^{16}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{17}{\left(x \right)}}{17}$
      
      So, the result is: $- \frac{165 \cot^{17}{\left(x \right)}}{17}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 330 \cot^{14}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 330 \int \cot^{14}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{15}{\left(x \right)}}{15}$
      
      So, the result is: $- 22 \cot^{15}{\left(x \right)}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 462 \cot^{12}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 462 \int \cot^{12}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{13}{\left(x \right)}}{13}$
      
      So, the result is: $- \frac{462 \cot^{13}{\left(x \right)}}{13}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 462 \cot^{10}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 462 \int \cot^{10}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{11}{\left(x \right)}}{11}$
      
      So, the result is: $- 42 \cot^{11}{\left(x \right)}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 330 \cot^{8}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 330 \int \cot^{8}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{9}{\left(x \right)}}{9}$
      
      So, the result is: $- \frac{110 \cot^{9}{\left(x \right)}}{3}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 165 \cot^{6}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 165 \int \cot^{6}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{7}{\left(x \right)}}{7}$
      
      So, the result is: $- \frac{165 \cot^{7}{\left(x \right)}}{7}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 55 \cot^{4}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 55 \int \cot^{4}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{5}{\left(x \right)}}{5}$
      
      So, the result is: $- 11 \cot^{5}{\left(x \right)}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 11 \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 11 \int \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{3}{\left(x \right)}}{3}$
      
      So, the result is: $- \frac{11 \cot^{3}{\left(x \right)}}{3}$
      
      $\int \csc^{2}{\left(x \right)}\, dx = - \cot{\left(x \right)}$
      
      The result is: $- \frac{\cot^{23}{\left(x \right)}}{23} - \frac{11 \cot^{21}{\left(x \right)}}{21} - \frac{55 \cot^{19}{\left(x \right)}}{19} - \frac{165 \cot^{17}{\left(x \right)}}{17} - 22 \cot^{15}{\left(x \right)} - \frac{462 \cot^{13}{\left(x \right)}}{13} - 42 \cot^{11}{\left(x \right)} - \frac{110 \cot^{9}{\left(x \right)}}{3} - \frac{165 \cot^{7}{\left(x \right)}}{7} - 11 \cot^{5}{\left(x \right)} - \frac{11 \cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}$
      Method #2
      
      Rewrite the integrand:
      
      $\left(\cot^{2}{\left(x \right)} + 1\right)^{11} \csc^{2}{\left(x \right)} = \cot^{22}{\left(x \right)} \csc^{2}{\left(x \right)} + 11 \cot^{20}{\left(x \right)} \csc^{2}{\left(x \right)} + 55 \cot^{18}{\left(x \right)} \csc^{2}{\left(x \right)} + 165 \cot^{16}{\left(x \right)} \csc^{2}{\left(x \right)} + 330 \cot^{14}{\left(x \right)} \csc^{2}{\left(x \right)} + 462 \cot^{12}{\left(x \right)} \csc^{2}{\left(x \right)} + 462 \cot^{10}{\left(x \right)} \csc^{2}{\left(x \right)} + 330 \cot^{8}{\left(x \right)} \csc^{2}{\left(x \right)} + 165 \cot^{6}{\left(x \right)} \csc^{2}{\left(x \right)} + 55 \cot^{4}{\left(x \right)} \csc^{2}{\left(x \right)} + 11 \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)} + \csc^{2}{\left(x \right)}$
      
      Integrate term-by-term:
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{23}{\left(x \right)}}{23}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 11 \cot^{20}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 11 \int \cot^{20}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{21}{\left(x \right)}}{21}$
      
      So, the result is: $- \frac{11 \cot^{21}{\left(x \right)}}{21}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 55 \cot^{18}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 55 \int \cot^{18}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{19}{\left(x \right)}}{19}$
      
      So, the result is: $- \frac{55 \cot^{19}{\left(x \right)}}{19}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 165 \cot^{16}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 165 \int \cot^{16}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{17}{\left(x \right)}}{17}$
      
      So, the result is: $- \frac{165 \cot^{17}{\left(x \right)}}{17}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 330 \cot^{14}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 330 \int \cot^{14}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{15}{\left(x \right)}}{15}$
      
      So, the result is: $- 22 \cot^{15}{\left(x \right)}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 462 \cot^{12}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 462 \int \cot^{12}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{13}{\left(x \right)}}{13}$
      
      So, the result is: $- \frac{462 \cot^{13}{\left(x \right)}}{13}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 462 \cot^{10}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 462 \int \cot^{10}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{11}{\left(x \right)}}{11}$
      
      So, the result is: $- 42 \cot^{11}{\left(x \right)}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 330 \cot^{8}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 330 \int \cot^{8}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{9}{\left(x \right)}}{9}$
      
      So, the result is: $- \frac{110 \cot^{9}{\left(x \right)}}{3}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 165 \cot^{6}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 165 \int \cot^{6}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{7}{\left(x \right)}}{7}$
      
      So, the result is: $- \frac{165 \cot^{7}{\left(x \right)}}{7}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 55 \cot^{4}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 55 \int \cot^{4}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{5}{\left(x \right)}}{5}$
      
      So, the result is: $- 11 \cot^{5}{\left(x \right)}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 11 \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 11 \int \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$
      
      Let $u = \cot{\left(x \right)}$ .
      
      Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ 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:
      
      $- \frac{\cot^{3}{\left(x \right)}}{3}$
      
      So, the result is: $- \frac{11 \cot^{3}{\left(x \right)}}{3}$
      
      $\int \csc^{2}{\left(x \right)}\, dx = - \cot{\left(x \right)}$
      
      The result is: $- \frac{\cot^{23}{\left(x \right)}}{23} - \frac{11 \cot^{21}{\left(x \right)}}{21} - \frac{55 \cot^{19}{\left(x \right)}}{19} - \frac{165 \cot^{17}{\left(x \right)}}{17} - 22 \cot^{15}{\left(x \right)} - \frac{462 \cot^{13}{\left(x \right)}}{13} - 42 \cot^{11}{\left(x \right)} - \frac{110 \cot^{9}{\left(x \right)}}{3} - \frac{165 \cot^{7}{\left(x \right)}}{7} - 11 \cot^{5}{\left(x \right)} - \frac{11 \cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}$
    So, the result is: $\frac{3 \cot^{23}{\left(x \right)}}{23} + \frac{11 \cot^{21}{\left(x \right)}}{7} + \frac{165 \cot^{19}{\left(x \right)}}{19} + \frac{495 \cot^{17}{\left(x \right)}}{17} + 66 \cot^{15}{\left(x \right)} + \frac{1386 \cot^{13}{\left(x \right)}}{13} + 126 \cot^{11}{\left(x \right)} + 110 \cot^{9}{\left(x \right)} + \frac{495 \cot^{7}{\left(x \right)}}{7} + 33 \cot^{5}{\left(x \right)} + 11 \cot^{3}{\left(x \right)} + 3 \cot{\left(x \right)}$
  The result is: $\frac{4 x^{6}}{3} + \frac{3 \cot^{23}{\left(x \right)}}{23} + \frac{11 \cot^{21}{\left(x \right)}}{7} + \frac{165 \cot^{19}{\left(x \right)}}{19} + \frac{495 \cot^{17}{\left(x \right)}}{17} + 66 \cot^{15}{\left(x \right)} + \frac{1386 \cot^{13}{\left(x \right)}}{13} + 126 \cot^{11}{\left(x \right)} + 110 \cot^{9}{\left(x \right)} + \frac{495 \cot^{7}{\left(x \right)}}{7} + 33 \cot^{5}{\left(x \right)} + 11 \cot^{3}{\left(x \right)} + 3 \cot{\left(x \right)}$
The result is: $\frac{4 x^{6}}{3} + \frac{2 x^{3}}{9} + \frac{3 \cot^{23}{\left(x \right)}}{23} + \frac{11 \cot^{21}{\left(x \right)}}{7} + \frac{165 \cot^{19}{\left(x \right)}}{19} + \frac{495 \cot^{17}{\left(x \right)}}{17} + 66 \cot^{15}{\left(x \right)} + \frac{1386 \cot^{13}{\left(x \right)}}{13} + 126 \cot^{11}{\left(x \right)} + 110 \cot^{9}{\left(x \right)} + \frac{495 \cot^{7}{\left(x \right)}}{7} + 33 \cot^{5}{\left(x \right)} + 11 \cot^{3}{\left(x \right)} + 3 \cot{\left(x \right)}$
Add the constant of integration:

$\frac{4 x^{6}}{3} + \frac{2 x^{3}}{9} + \frac{3 \cot^{23}{\left(x \right)}}{23} + \frac{11 \cot^{21}{\left(x \right)}}{7} + \frac{165 \cot^{19}{\left(x \right)}}{19} + \frac{495 \cot^{17}{\left(x \right)}}{17} + 66 \cot^{15}{\left(x \right)} + \frac{1386 \cot^{13}{\left(x \right)}}{13} + 126 \cot^{11}{\left(x \right)} + 110 \cot^{9}{\left(x \right)} + \frac{495 \cot^{7}{\left(x \right)}}{7} + 33 \cot^{5}{\left(x \right)} + 11 \cot^{3}{\left(x \right)} + 3 \cot{\left(x \right)}+ \mathrm{constant}$

The answer is:

$\frac{4 x^{6}}{3} + \frac{2 x^{3}}{9} + \frac{3 \cot^{23}{\left(x \right)}}{23} + \frac{11 \cot^{21}{\left(x \right)}}{7} + \frac{165 \cot^{19}{\left(x \right)}}{19} + \frac{495 \cot^{17}{\left(x \right)}}{17} + 66 \cot^{15}{\left(x \right)} + \frac{1386 \cot^{13}{\left(x \right)}}{13} + 126 \cot^{11}{\left(x \right)} + 110 \cot^{9}{\left(x \right)} + \frac{495 \cot^{7}{\left(x \right)}}{7} + 33 \cot^{5}{\left(x \right)} + 11 \cot^{3}{\left(x \right)} + 3 \cot{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                                                                                                                                    
 |                                                                                                                                                                                                                     
 | /                     2\                                                                                             3        23         6         21             19             7             17              13   
 | |   5      3       2*x |                           3            5            15             9             11      2*x    3*cot  (x)   4*x    11*cot  (x)   165*cot  (x)   495*cot (x)   495*cot  (x)   1386*cot  (x)
 | |8*x  - -------- + ----| dx = C + 3*cot(x) + 11*cot (x) + 33*cot (x) + 66*cot  (x) + 110*cot (x) + 126*cot  (x) + ---- + ---------- + ---- + ----------- + ------------ + ----------- + ------------ + -------------
 | |          24       3  |                                                                                           9         23        3          7             19             7             17              13     
 | \       sin  (x)       /                                                                                                                                                                                            
 |                                                                                                                                                                                                                     
/

$$\int \left(\frac{2 x^{2}}{3} + \left(8 x^{5} - \frac{3}{\sin^{24}{\left(x \right)}}\right)\right)\, dx = C + \frac{4 x^{6}}{3} + \frac{2 x^{3}}{9} + \frac{3 \cot^{23}{\left(x \right)}}{23} + \frac{11 \cot^{21}{\left(x \right)}}{7} + \frac{165 \cot^{19}{\left(x \right)}}{19} + \frac{495 \cot^{17}{\left(x \right)}}{17} + 66 \cot^{15}{\left(x \right)} + \frac{1386 \cot^{13}{\left(x \right)}}{13} + 126 \cot^{11}{\left(x \right)} + 110 \cot^{9}{\left(x \right)} + \frac{495 \cot^{7}{\left(x \right)}}{7} + 33 \cot^{5}{\left(x \right)} + 11 \cot^{3}{\left(x \right)} + 3 \cot{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-oo

$$-\infty$$

-oo

$$-\infty$$

-oo

Numerical answer [src]

-1.95714425086013e+437

-1.95714425086013e+437

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

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Identical expressions

Similar expressions

Integral of (8x⁵-3÷sin²4x+2÷3x²)dx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2