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Solving integrals/ cos(100x+2)dx

Integral of cos(100x+2)dx dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
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∫01cos⁡(100x+2) dx\int\limits_{0}^{1} \cos{\left(100 x + 2 \right)}\, dx 
Integral(cos(100*x + 2), (x, 0, 1))
Detail solution

  1. Let u=100x+2u = 100 x + 2.

    Then let du=100dxdu = 100 dx and substitute du100\frac{du}{100}:

    ∫cos⁡(u)100 du\int \frac{\cos{\left(u \right)}}{100}\, du

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

      ∫cos⁡(u) du=∫cos⁡(u) du100\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{100}

      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)100\frac{\sin{\left(u \right)}}{100}

    Now substitute uu back in:

    sin⁡(100x+2)100\frac{\sin{\left(100 x + 2 \right)}}{100}

  2. Now simplify:

    sin⁡(100x+2)100\frac{\sin{\left(100 x + 2 \right)}}{100}

  3. Add the constant of integration:

    sin⁡(100x+2)100+constant\frac{\sin{\left(100 x + 2 \right)}}{100}+ \mathrm{constant}

The answer is:

sin⁡(100x+2)100+constant\frac{\sin{\left(100 x + 2 \right)}}{100}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                      
 |                         sin(100*x + 2)
 | cos(100*x + 2) dx = C + --------------
 |                              100      
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∫cos⁡(100x+2) dx=C+sin⁡(100x+2)100\int \cos{\left(100 x + 2 \right)}\, dx = C + \frac{\sin{\left(100 x + 2 \right)}}{100}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
Plot the graph f(x)
The answer [src] 
  sin(2)   sin(102)
- ------ + --------
   100       100
−sin⁡(2)100+sin⁡(102)100- \frac{\sin{\left(2 \right)}}{100} + \frac{\sin{\left(102 \right)}}{100} 
=
= 
  sin(2)   sin(102)
- ------ + --------
   100       100
−sin⁡(2)100+sin⁡(102)100- \frac{\sin{\left(2 \right)}}{100} + \frac{\sin{\left(102 \right)}}{100} 
-sin(2)/100 + sin(102)/100
Numerical answer [src] 
0.000855293645327247
0.000855293645327247

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

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