Integral of sin10x*sin7x dx

The solution

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$$\int\limits_{0}^{0} \sin{\left(7 x \right)} \sin{\left(10 x \right)}\, dx$$

Integral(sin(10*x)*sin(7*x), (x, 0, 0))

Detail solution

Rewrite the integrand:

$\sin{\left(7 x \right)} \sin{\left(10 x \right)} = - 32768 \sin^{16}{\left(x \right)} \cos{\left(x \right)} + 122880 \sin^{14}{\left(x \right)} \cos{\left(x \right)} - 186368 \sin^{12}{\left(x \right)} \cos{\left(x \right)} + 146432 \sin^{10}{\left(x \right)} \cos{\left(x \right)} - 63360 \sin^{8}{\left(x \right)} \cos{\left(x \right)} + 14784 \sin^{6}{\left(x \right)} \cos{\left(x \right)} - 1680 \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 70 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 32768 \sin^{16}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 32768 \int \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx$
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{16}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{16}\, du = \frac{u^{17}}{17}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{17}{\left(x \right)}}{17}$
  So, the result is: $- \frac{32768 \sin^{17}{\left(x \right)}}{17}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 122880 \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx = 122880 \int \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx$
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{14}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{14}\, du = \frac{u^{15}}{15}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{15}{\left(x \right)}}{15}$
  So, the result is: $8192 \sin^{15}{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 186368 \sin^{12}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 186368 \int \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx$
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{12}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{12}\, du = \frac{u^{13}}{13}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{13}{\left(x \right)}}{13}$
  So, the result is: $- 14336 \sin^{13}{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 146432 \sin^{10}{\left(x \right)} \cos{\left(x \right)}\, dx = 146432 \int \sin^{10}{\left(x \right)} \cos{\left(x \right)}\, dx$
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{10}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{10}\, du = \frac{u^{11}}{11}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{11}{\left(x \right)}}{11}$
  So, the result is: $13312 \sin^{11}{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 63360 \sin^{8}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 63360 \int \sin^{8}{\left(x \right)} \cos{\left(x \right)}\, dx$
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{8}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{8}\, du = \frac{u^{9}}{9}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{9}{\left(x \right)}}{9}$
  So, the result is: $- 7040 \sin^{9}{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 14784 \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx = 14784 \int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx$
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{6}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{7}{\left(x \right)}}{7}$
  So, the result is: $2112 \sin^{7}{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 1680 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 1680 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{4}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{5}{\left(x \right)}}{5}$
  So, the result is: $- 336 \sin^{5}{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 70 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx = 70 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{2}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{3}{\left(x \right)}}{3}$
  So, the result is: $\frac{70 \sin^{3}{\left(x \right)}}{3}$
The result is: $- \frac{32768 \sin^{17}{\left(x \right)}}{17} + 8192 \sin^{15}{\left(x \right)} - 14336 \sin^{13}{\left(x \right)} + 13312 \sin^{11}{\left(x \right)} - 7040 \sin^{9}{\left(x \right)} + 2112 \sin^{7}{\left(x \right)} - 336 \sin^{5}{\left(x \right)} + \frac{70 \sin^{3}{\left(x \right)}}{3}$
Now simplify:

$\frac{2 \left(- 49152 \sin^{14}{\left(x \right)} + 208896 \sin^{12}{\left(x \right)} - 365568 \sin^{10}{\left(x \right)} + 339456 \sin^{8}{\left(x \right)} - 179520 \sin^{6}{\left(x \right)} + 53856 \sin^{4}{\left(x \right)} - 8568 \sin^{2}{\left(x \right)} + 595\right) \sin^{3}{\left(x \right)}}{51}$
Add the constant of integration:

$\frac{2 \left(- 49152 \sin^{14}{\left(x \right)} + 208896 \sin^{12}{\left(x \right)} - 365568 \sin^{10}{\left(x \right)} + 339456 \sin^{8}{\left(x \right)} - 179520 \sin^{6}{\left(x \right)} + 53856 \sin^{4}{\left(x \right)} - 8568 \sin^{2}{\left(x \right)} + 595\right) \sin^{3}{\left(x \right)}}{51}+ \mathrm{constant}$

The answer is:

$\frac{2 \left(- 49152 \sin^{14}{\left(x \right)} + 208896 \sin^{12}{\left(x \right)} - 365568 \sin^{10}{\left(x \right)} + 339456 \sin^{8}{\left(x \right)} - 179520 \sin^{6}{\left(x \right)} + 53856 \sin^{4}{\left(x \right)} - 8568 \sin^{2}{\left(x \right)} + 595\right) \sin^{3}{\left(x \right)}}{51}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                                                   17            3   
 |                                      13              9             5              7              15               11      32768*sin  (x)   70*sin (x)
 | sin(10*x)*sin(7*x) dx = C - 14336*sin  (x) - 7040*sin (x) - 336*sin (x) + 2112*sin (x) + 8192*sin  (x) + 13312*sin  (x) - -------------- + ----------
 |                                                                                                                                 17             3     
/

$$\int \sin{\left(7 x \right)} \sin{\left(10 x \right)}\, dx = C - \frac{32768 \sin^{17}{\left(x \right)}}{17} + 8192 \sin^{15}{\left(x \right)} - 14336 \sin^{13}{\left(x \right)} + 13312 \sin^{11}{\left(x \right)} - 7040 \sin^{9}{\left(x \right)} + 2112 \sin^{7}{\left(x \right)} - 336 \sin^{5}{\left(x \right)} + \frac{70 \sin^{3}{\left(x \right)}}{3}$$

Simplify

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Numerical answer [src]

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

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Integral of sin10x*sin7x dx

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