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sin(4x-1)

Integral of sin(4x-1) dx

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The solution

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 |  sin(4*x - 1) dx
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01sin(4x1)dx\int\limits_{0}^{1} \sin{\left(4 x - 1 \right)}\, dx
Integral(sin(4*x - 1), (x, 0, 1))
Detail solution
  1. Let u=4x1u = 4 x - 1.

    Then let du=4dxdu = 4 dx and substitute du4\frac{du}{4}:

    sin(u)4du\int \frac{\sin{\left(u \right)}}{4}\, du

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

      sin(u)du=sin(u)du4\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}

      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)4- \frac{\cos{\left(u \right)}}{4}

    Now substitute uu back in:

    cos(4x1)4- \frac{\cos{\left(4 x - 1 \right)}}{4}

  2. Now simplify:

    cos(4x1)4- \frac{\cos{\left(4 x - 1 \right)}}{4}

  3. Add the constant of integration:

    cos(4x1)4+constant- \frac{\cos{\left(4 x - 1 \right)}}{4}+ \mathrm{constant}


The answer is:

cos(4x1)4+constant- \frac{\cos{\left(4 x - 1 \right)}}{4}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                  
 |                       cos(4*x - 1)
 | sin(4*x - 1) dx = C - ------------
 |                            4      
/                                    
sin(4x1)dx=Ccos(4x1)4\int \sin{\left(4 x - 1 \right)}\, dx = C - \frac{\cos{\left(4 x - 1 \right)}}{4}
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
The answer [src]
  cos(3)   cos(1)
- ------ + ------
    4        4   
cos(1)4cos(3)4\frac{\cos{\left(1 \right)}}{4} - \frac{\cos{\left(3 \right)}}{4}
=
=
  cos(3)   cos(1)
- ------ + ------
    4        4   
cos(1)4cos(3)4\frac{\cos{\left(1 \right)}}{4} - \frac{\cos{\left(3 \right)}}{4}
-cos(3)/4 + cos(1)/4
Numerical answer [src]
0.382573700617146
0.382573700617146
The graph
Integral of sin(4x-1) dx

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