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Solving integrals/ cos(2x−5)

Integral of cos(2x−5) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
 52                
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 19                
  /                
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 |  cos(2*x - 5) dx
 |                 
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103                
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 38
∫103385219cos⁡(2x−5) dx\int\limits_{\frac{103}{38}}^{\frac{52}{19}} \cos{\left(2 x - 5 \right)}\, dx 
Integral(cos(2*x - 5), (x, 103/38, 52/19))
Detail solution

  1. Let u=2x−5u = 2 x - 5.

    Then let du=2dxdu = 2 dx and substitute du2\frac{du}{2}:

    ∫cos⁡(u)2 du\int \frac{\cos{\left(u \right)}}{2}\, du

    1. The integral of a constant times a function is the constant times the integral of the function:

      ∫cos⁡(u) du=∫cos⁡(u) du2\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}

      1. The integral of cosine is sine:

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

      So, the result is: sin⁡(u)2\frac{\sin{\left(u \right)}}{2}

    Now substitute uu back in:

    sin⁡(2x−5)2\frac{\sin{\left(2 x - 5 \right)}}{2}

  2. Now simplify:

    sin⁡(2x−5)2\frac{\sin{\left(2 x - 5 \right)}}{2}

  3. Add the constant of integration:

    sin⁡(2x−5)2+constant\frac{\sin{\left(2 x - 5 \right)}}{2}+ \mathrm{constant}

The answer is:

sin⁡(2x−5)2+constant\frac{\sin{\left(2 x - 5 \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                  
 |                       sin(2*x - 5)
 | cos(2*x - 5) dx = C + ------------
 |                            2      
/
∫cos⁡(2x−5) dx=C+sin⁡(2x−5)2\int \cos{\left(2 x - 5 \right)}\, dx = C + \frac{\sin{\left(2 x - 5 \right)}}{2}
Simplify
The graph
2.71252.71502.71752.72002.72252.72502.72752.73002.73252.73500.01.0
Plot the graph f(x)
The answer [src] 
sin(9/19)   sin(8/19)
--------- - ---------
    2           2
−sin⁡(819)2+sin⁡(919)2- \frac{\sin{\left(\frac{8}{19} \right)}}{2} + \frac{\sin{\left(\frac{9}{19} \right)}}{2} 
=
= 
sin(9/19)   sin(8/19)
--------- - ---------
    2           2
−sin⁡(819)2+sin⁡(919)2- \frac{\sin{\left(\frac{8}{19} \right)}}{2} + \frac{\sin{\left(\frac{9}{19} \right)}}{2} 
sin(9/19)/2 - sin(8/19)/2
Numerical answer [src] 
0.0237232781950296
0.0237232781950296

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

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