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cos(2pix)cos(pix)

Integral of cos(2pix)cos(pix) dx

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 |  cos(2*pi*x)*cos(pi*x) dx
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01cos(πx)cos(2πx)dx\int\limits_{0}^{1} \cos{\left(\pi x \right)} \cos{\left(2 \pi x \right)}\, dx
Integral(cos(2*pi*x)*cos(pi*x), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=πxu = \pi x.

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

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

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

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

        1. Rewrite the integrand:

          cos(2u)cos(u)=2cos3(u)cos(u)\cos{\left(2 u \right)} \cos{\left(u \right)} = 2 \cos^{3}{\left(u \right)} - \cos{\left(u \right)}

        2. Integrate term-by-term:

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

            2cos3(u)du=2cos3(u)du\int 2 \cos^{3}{\left(u \right)}\, du = 2 \int \cos^{3}{\left(u \right)}\, du

            1. Rewrite the integrand:

              cos3(u)=(1sin2(u))cos(u)\cos^{3}{\left(u \right)} = \left(1 - \sin^{2}{\left(u \right)}\right) \cos{\left(u \right)}

            2. Let u=sin(u)u = \sin{\left(u \right)}.

              Then let du=cos(u)dudu = \cos{\left(u \right)} du and substitute dudu:

              (1u2)du\int \left(1 - u^{2}\right)\, du

              1. Integrate term-by-term:

                1. The integral of a constant is the constant times the variable of integration:

                  1du=u\int 1\, du = u

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

                  (u2)du=u2du\int \left(- u^{2}\right)\, du = - \int u^{2}\, du

                  1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

                    u2du=u33\int u^{2}\, du = \frac{u^{3}}{3}

                  So, the result is: u33- \frac{u^{3}}{3}

                The result is: u33+u- \frac{u^{3}}{3} + u

              Now substitute uu back in:

              sin3(u)3+sin(u)- \frac{\sin^{3}{\left(u \right)}}{3} + \sin{\left(u \right)}

            So, the result is: 2sin3(u)3+2sin(u)- \frac{2 \sin^{3}{\left(u \right)}}{3} + 2 \sin{\left(u \right)}

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

            (cos(u))du=cos(u)du\int \left(- \cos{\left(u \right)}\right)\, du = - \int \cos{\left(u \right)}\, du

            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)- \sin{\left(u \right)}

          The result is: 2sin3(u)3+sin(u)- \frac{2 \sin^{3}{\left(u \right)}}{3} + \sin{\left(u \right)}

        So, the result is: 2sin3(u)3+sin(u)π\frac{- \frac{2 \sin^{3}{\left(u \right)}}{3} + \sin{\left(u \right)}}{\pi}

      Now substitute uu back in:

      2sin3(πx)3+sin(πx)π\frac{- \frac{2 \sin^{3}{\left(\pi x \right)}}{3} + \sin{\left(\pi x \right)}}{\pi}

    Method #2

    1. Rewrite the integrand:

      cos(πx)cos(2πx)=2cos3(πx)cos(πx)\cos{\left(\pi x \right)} \cos{\left(2 \pi x \right)} = 2 \cos^{3}{\left(\pi x \right)} - \cos{\left(\pi x \right)}

    2. Integrate term-by-term:

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

        2cos3(πx)dx=2cos3(πx)dx\int 2 \cos^{3}{\left(\pi x \right)}\, dx = 2 \int \cos^{3}{\left(\pi x \right)}\, dx

        1. Rewrite the integrand:

          cos3(πx)=(1sin2(πx))cos(πx)\cos^{3}{\left(\pi x \right)} = \left(1 - \sin^{2}{\left(\pi x \right)}\right) \cos{\left(\pi x \right)}

        2. Let u=sin(πx)u = \sin{\left(\pi x \right)}.

          Then let du=πcos(πx)dxdu = \pi \cos{\left(\pi x \right)} dx and substitute duπ\frac{du}{\pi}:

          1u2π2du\int \frac{1 - u^{2}}{\pi^{2}}\, du

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

            1u2πdu=(1u2)duπ\int \frac{1 - u^{2}}{\pi}\, du = \frac{\int \left(1 - u^{2}\right)\, du}{\pi}

            1. Integrate term-by-term:

              1. The integral of a constant is the constant times the variable of integration:

                1du=u\int 1\, du = u

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

                (u2)du=u2du\int \left(- u^{2}\right)\, du = - \int u^{2}\, du

                1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

                  u2du=u33\int u^{2}\, du = \frac{u^{3}}{3}

                So, the result is: u33- \frac{u^{3}}{3}

              The result is: u33+u- \frac{u^{3}}{3} + u

            So, the result is: u33+uπ\frac{- \frac{u^{3}}{3} + u}{\pi}

          Now substitute uu back in:

          sin3(πx)3+sin(πx)π\frac{- \frac{\sin^{3}{\left(\pi x \right)}}{3} + \sin{\left(\pi x \right)}}{\pi}

        So, the result is: 2(sin3(πx)3+sin(πx))π\frac{2 \left(- \frac{\sin^{3}{\left(\pi x \right)}}{3} + \sin{\left(\pi x \right)}\right)}{\pi}

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

        (cos(πx))dx=cos(πx)dx\int \left(- \cos{\left(\pi x \right)}\right)\, dx = - \int \cos{\left(\pi x \right)}\, dx

        1. Let u=πxu = \pi x.

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

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

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

            cos(u)πdu=cos(u)duπ\int \frac{\cos{\left(u \right)}}{\pi}\, du = \frac{\int \cos{\left(u \right)}\, du}{\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)π\frac{\sin{\left(u \right)}}{\pi}

          Now substitute uu back in:

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

        So, the result is: sin(πx)π- \frac{\sin{\left(\pi x \right)}}{\pi}

      The result is: 2(sin3(πx)3+sin(πx))πsin(πx)π\frac{2 \left(- \frac{\sin^{3}{\left(\pi x \right)}}{3} + \sin{\left(\pi x \right)}\right)}{\pi} - \frac{\sin{\left(\pi x \right)}}{\pi}

  2. Now simplify:

    (cos(2πx)+2)sin(πx)3π\frac{\left(\cos{\left(2 \pi x \right)} + 2\right) \sin{\left(\pi x \right)}}{3 \pi}

  3. Add the constant of integration:

    (cos(2πx)+2)sin(πx)3π+constant\frac{\left(\cos{\left(2 \pi x \right)} + 2\right) \sin{\left(\pi x \right)}}{3 \pi}+ \mathrm{constant}


The answer is:

(cos(2πx)+2)sin(πx)3π+constant\frac{\left(\cos{\left(2 \pi x \right)} + 2\right) \sin{\left(\pi x \right)}}{3 \pi}+ \mathrm{constant}

The answer (Indefinite) [src]
                                         3                  
                                    2*sin (pi*x)            
  /                               - ------------ + sin(pi*x)
 |                                       3                  
 | cos(2*pi*x)*cos(pi*x) dx = C + --------------------------
 |                                            pi            
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sin(3πx)6π+sin(πx)2π{{\sin \left(3\,\pi\,x\right)}\over{6\,\pi}}+{{\sin \left(\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(3π)+3sinπ6π{{\sin \left(3\,\pi\right)+3\,\sin \pi}\over{6\,\pi}}
=
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0
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Numerical answer [src]
-1.02704285974057e-22
-1.02704285974057e-22
The graph
Integral of cos(2pix)cos(pix) dx

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