Mister Exam

Integral of cos(2pi(x)) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
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 |  cos(2*pi*x) dx
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01cos(2πx)dx\int\limits_{0}^{1} \cos{\left(2 \pi x \right)}\, dx
Detail solution
  1. Let u=2πxu = 2 \pi x.

    Then let du=2πdxdu = 2 \pi dx and substitute du2π\frac{du}{2 \pi}:

    cos(u)4π2du\int \frac{\cos{\left(u \right)}}{4 \pi^{2}}\, du

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

      cos(u)2πdu=cos(u)du2π\int \frac{\cos{\left(u \right)}}{2 \pi}\, du = \frac{\int \cos{\left(u \right)}\, du}{2 \pi}

      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)2π\frac{\sin{\left(u \right)}}{2 \pi}

    Now substitute uu back in:

    sin(2πx)2π\frac{\sin{\left(2 \pi x \right)}}{2 \pi}

  2. Add the constant of integration:

    sin(2πx)2π+constant\frac{\sin{\left(2 \pi x \right)}}{2 \pi}+ \mathrm{constant}


The answer is:

sin(2πx)2π+constant\frac{\sin{\left(2 \pi x \right)}}{2 \pi}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                
 |                      sin(2*pi*x)
 | cos(2*pi*x) dx = C + -----------
 |                          2*pi   
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sin(2πx)2π{{\sin \left(2\,\pi\,x\right)}\over{2\,\pi}}
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
The answer [src]
0
sin(2π)2π{{\sin \left(2\,\pi\right)}\over{2\,\pi}}
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=
0
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Numerical answer [src]
-7.41845798679675e-20
-7.41845798679675e-20
The graph
Integral of cos(2pi(x)) dx

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