Integral of sin²(5x) dx

The solution

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 5*l            
 ---            
  6             
  /             
 |              
 |     2        
 |  sin (5*x) dx
 |              
/               
l               
-               
6

$$\int\limits_{\frac{l}{6}}^{\frac{5 l}{6}} \sin^{2}{\left(5 x \right)}\, dx$$

Integral(sin(5*x)^2, (x, l/6, 5*l/6))

Detail solution

Rewrite the integrand:

$\sin^{2}{\left(5 x \right)} = \frac{1}{2} - \frac{\cos{\left(10 x \right)}}{2}$
Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \frac{1}{2}\, dx = \frac{x}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\cos{\left(10 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(10 x \right)}\, dx}{2}$
  1. Let $u = 10 x$ .
    
    Then let $du = 10 dx$ and substitute $\frac{du}{10}$ :
    
    $\int \frac{\cos{\left(u \right)}}{10}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{10}$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      So, the result is: $\frac{\sin{\left(u \right)}}{10}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(10 x \right)}}{10}$
  So, the result is: $- \frac{\sin{\left(10 x \right)}}{20}$
The result is: $\frac{x}{2} - \frac{\sin{\left(10 x \right)}}{20}$
Add the constant of integration:

$\frac{x}{2} - \frac{\sin{\left(10 x \right)}}{20}+ \mathrm{constant}$

The answer is:

$\frac{x}{2} - \frac{\sin{\left(10 x \right)}}{20}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                
 |                                 
 |    2               x   sin(10*x)
 | sin (5*x) dx = C + - - ---------
 |                    2       20   
/

$$\int \sin^{2}{\left(5 x \right)}\, dx = C + \frac{x}{2} - \frac{\sin{\left(10 x \right)}}{20}$$

Simplify

The answer [src]

       /25*l\    /25*l\      /5*l\    /5*l\
    cos|----|*sin|----|   cos|---|*sin|---|
l      \ 6  /    \ 6  /      \ 6 /    \ 6 /
- - ------------------- + -----------------
3            10                   10

$$\frac{l}{3} + \frac{\sin{\left(\frac{5 l}{6} \right)} \cos{\left(\frac{5 l}{6} \right)}}{10} - \frac{\sin{\left(\frac{25 l}{6} \right)} \cos{\left(\frac{25 l}{6} \right)}}{10}$$

       /25*l\    /25*l\      /5*l\    /5*l\
    cos|----|*sin|----|   cos|---|*sin|---|
l      \ 6  /    \ 6  /      \ 6 /    \ 6 /
- - ------------------- + -----------------
3            10                   10

$$\frac{l}{3} + \frac{\sin{\left(\frac{5 l}{6} \right)} \cos{\left(\frac{5 l}{6} \right)}}{10} - \frac{\sin{\left(\frac{25 l}{6} \right)} \cos{\left(\frac{25 l}{6} \right)}}{10}$$

l/3 - cos(25*l/6)*sin(25*l/6)/10 + cos(5*l/6)*sin(5*l/6)/10

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

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Integral of sin²(5x) dx

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