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Solving integrals/ (-2/3)(sen(4x)dx)

Integral of (-2/3)(sen(4x)dx) dx

Limits of integration:

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The solution

You have entered [src] 
  1               
  /               
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 |  -2*sin(4*x)   
 |  ----------- dx
 |       3        
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0
∫01(−2sin⁡(4x)3) dx\int\limits_{0}^{1} \left(- \frac{2 \sin{\left(4 x \right)}}{3}\right)\, dx 
Integral(-2*sin(4*x)/3, (x, 0, 1))
Detail solution

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

    ∫(−2sin⁡(4x)3) dx=−2∫sin⁡(4x) dx3\int \left(- \frac{2 \sin{\left(4 x \right)}}{3}\right)\, dx = - \frac{2 \int \sin{\left(4 x \right)}\, dx}{3}

    1. Let u=4xu = 4 x.

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

      ∫sin⁡(u)4 du\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⁡(4x)4- \frac{\cos{\left(4 x \right)}}{4}

    So, the result is: cos⁡(4x)6\frac{\cos{\left(4 x \right)}}{6}

  2. Add the constant of integration:

    cos⁡(4x)6+constant\frac{\cos{\left(4 x \right)}}{6}+ \mathrm{constant}

The answer is:

cos⁡(4x)6+constant\frac{\cos{\left(4 x \right)}}{6}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                             
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 | -2*sin(4*x)          cos(4*x)
 | ----------- dx = C + --------
 |      3                  6    
 |                              
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∫(−2sin⁡(4x)3) dx=C+cos⁡(4x)6\int \left(- \frac{2 \sin{\left(4 x \right)}}{3}\right)\, dx = C + \frac{\cos{\left(4 x \right)}}{6}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901-1
Plot the graph f(x)
The answer [src] 
  1   cos(4)
- - + ------
  6     6
−16+cos⁡(4)6- \frac{1}{6} + \frac{\cos{\left(4 \right)}}{6} 
=
= 
  1   cos(4)
- - + ------
  6     6
−16+cos⁡(4)6- \frac{1}{6} + \frac{\cos{\left(4 \right)}}{6} 
-1/6 + cos(4)/6
Numerical answer [src] 
-0.275607270143935
-0.275607270143935

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

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