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Solving integrals/ (sin5x-sin5a)

Integral of (sin5x-sin5a) dx

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The solution

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 |  (sin(5*x) - sin(5*a)) dx
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∫01(−sin⁡(5a)+sin⁡(5x)) dx\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

  1. Integrate term-by-term:

    1. The integral of a constant is the constant times the variable of integration:

      ∫(−sin⁡(5a)) dx=−xsin⁡(5a)\int \left(- \sin{\left(5 a \right)}\right)\, dx = - x \sin{\left(5 a \right)}

    1. Let u=5xu = 5 x.

      Then let du=5dxdu = 5 dx and substitute du5\frac{du}{5}:

      ∫sin⁡(u)5 du\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:

        ∫sin⁡(u) du=∫sin⁡(u) du5\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}

        1. The integral of sine is negative cosine:

          ∫sin⁡(u) du=−cos⁡(u)\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}

        So, the result is: −cos⁡(u)5- \frac{\cos{\left(u \right)}}{5}

      Now substitute uu back in:

      −cos⁡(5x)5- \frac{\cos{\left(5 x \right)}}{5}

    The result is: −xsin⁡(5a)−cos⁡(5x)5- x \sin{\left(5 a \right)} - \frac{\cos{\left(5 x \right)}}{5}

  2. Add the constant of integration:

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

The answer is:

−xsin⁡(5a)−cos⁡(5x)5+constant- 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                 
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∫(−sin⁡(5a)+sin⁡(5x)) dx=C−xsin⁡(5a)−cos⁡(5x)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⁡(5a)−cos⁡(5)5+15- \sin{\left(5 a \right)} - \frac{\cos{\left(5 \right)}}{5} + \frac{1}{5} 
=
= 
1              cos(5)
- - sin(5*a) - ------
5                5
−sin⁡(5a)−cos⁡(5)5+15- \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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