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Solving integrals/ cos(3x-5)

Integral of cos(3x-5) dx

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The solution

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  0                
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 |  cos(3*x - 5) dx
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-oo
∫−∞0cos⁡(3x−5) dx\int\limits_{-\infty}^{0} \cos{\left(3 x - 5 \right)}\, dx 
Integral(cos(3*x - 5), (x, -oo, 0))
Detail solution

  1. Let u=3x−5u = 3 x - 5.

    Then let du=3dxdu = 3 dx and substitute du3\frac{du}{3}:

    ∫cos⁡(u)3 du\int \frac{\cos{\left(u \right)}}{3}\, du

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

      ∫cos⁡(u) du=∫cos⁡(u) du3\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{3}

      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)3\frac{\sin{\left(u \right)}}{3}

    Now substitute uu back in:

    sin⁡(3x−5)3\frac{\sin{\left(3 x - 5 \right)}}{3}

  2. Now simplify:

    sin⁡(3x−5)3\frac{\sin{\left(3 x - 5 \right)}}{3}

  3. Add the constant of integration:

    sin⁡(3x−5)3+constant\frac{\sin{\left(3 x - 5 \right)}}{3}+ \mathrm{constant}

The answer is:

sin⁡(3x−5)3+constant\frac{\sin{\left(3 x - 5 \right)}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                  
 |                       sin(3*x - 5)
 | cos(3*x - 5) dx = C + ------------
 |                            3      
/
∫cos⁡(3x−5) dx=C+sin⁡(3x−5)3\int \cos{\left(3 x - 5 \right)}\, dx = C + \frac{\sin{\left(3 x - 5 \right)}}{3}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.250.35
Plot the graph f(x)
The answer [src] 
   1   sin(5)  1   sin(5) 
<- - - ------, - - ------>
   3     3     3     3
⟨−13−sin⁡(5)3,13−sin⁡(5)3⟩\left\langle - \frac{1}{3} - \frac{\sin{\left(5 \right)}}{3}, \frac{1}{3} - \frac{\sin{\left(5 \right)}}{3}\right\rangle 
=
= 
   1   sin(5)  1   sin(5) 
<- - - ------, - - ------>
   3     3     3     3
⟨−13−sin⁡(5)3,13−sin⁡(5)3⟩\left\langle - \frac{1}{3} - \frac{\sin{\left(5 \right)}}{3}, \frac{1}{3} - \frac{\sin{\left(5 \right)}}{3}\right\rangle 
AccumBounds(-1/3 - sin(5)/3, 1/3 - sin(5)/3)

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

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