Integral of (sin5x-sin5a) dx

The solution

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  1                         
  /                         
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 |  (sin(5*x) - sin(5*a)) dx
 |                          
/                           
0

$$\int\limits_{0}^{1} \left(- \sin{\left(5 a \right)} + \sin{\left(5 x \right)}\right)\, dx$$

Integral(sin(5*x) - sin(5*a), (x, 0, 1))

Detail solution

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

$- x \sin{\left(5 a \right)} - \frac{\cos{\left(5 x \right)}}{5}+ \mathrm{constant}$

The answer is:

$- x \sin{\left(5 a \right)} - \frac{\cos{\left(5 x \right)}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                    
 |                                cos(5*x)             
 | (sin(5*x) - sin(5*a)) dx = C - -------- - x*sin(5*a)
 |                                   5                 
/

$$\int \left(- \sin{\left(5 a \right)} + \sin{\left(5 x \right)}\right)\, dx = C - x \sin{\left(5 a \right)} - \frac{\cos{\left(5 x \right)}}{5}$$

Simplify

The answer [src]

1              cos(5)
- - sin(5*a) - ------
5                5

$$- \sin{\left(5 a \right)} - \frac{\cos{\left(5 \right)}}{5} + \frac{1}{5}$$

1              cos(5)
- - sin(5*a) - ------
5                5

$$- \sin{\left(5 a \right)} - \frac{\cos{\left(5 \right)}}{5} + \frac{1}{5}$$

1/5 - sin(5*a) - cos(5)/5

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

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Integral of (sin5x-sin5a) dx

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