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sin(x/2)cos(x/2)

Integral of sin(x/2)cos(x/2) dx

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The solution

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01sin(x2)cos(x2)dx\int\limits_{0}^{1} \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}\, dx
Integral(sin(x/2)*cos(x/2), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=sin(x2)u = \sin{\left(\frac{x}{2} \right)}.

      Then let du=cos(x2)dx2du = \frac{\cos{\left(\frac{x}{2} \right)} dx}{2} and substitute 2du2 du:

      2udu\int 2 u\, du

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

        udu=2udu\int u\, du = 2 \int u\, du

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

          udu=u22\int u\, du = \frac{u^{2}}{2}

        So, the result is: u2u^{2}

      Now substitute uu back in:

      sin2(x2)\sin^{2}{\left(\frac{x}{2} \right)}

    Method #2

    1. Let u=x2u = \frac{x}{2}.

      Then let du=dx2du = \frac{dx}{2} and substitute 2du2 du:

      2sin(u)cos(u)du\int 2 \sin{\left(u \right)} \cos{\left(u \right)}\, du

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

        sin(u)cos(u)du=2sin(u)cos(u)du\int \sin{\left(u \right)} \cos{\left(u \right)}\, du = 2 \int \sin{\left(u \right)} \cos{\left(u \right)}\, du

        1. Let u=cos(u)u = \cos{\left(u \right)}.

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

          (u)du\int \left(- u\right)\, du

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

            udu=udu\int u\, du = - \int u\, du

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

              udu=u22\int u\, du = \frac{u^{2}}{2}

            So, the result is: u22- \frac{u^{2}}{2}

          Now substitute uu back in:

          cos2(u)2- \frac{\cos^{2}{\left(u \right)}}{2}

        So, the result is: cos2(u)- \cos^{2}{\left(u \right)}

      Now substitute uu back in:

      cos2(x2)- \cos^{2}{\left(\frac{x}{2} \right)}

    Method #3

    1. Let u=cos(x2)u = \cos{\left(\frac{x}{2} \right)}.

      Then let du=sin(x2)dx2du = - \frac{\sin{\left(\frac{x}{2} \right)} dx}{2} and substitute 2du- 2 du:

      (2u)du\int \left(- 2 u\right)\, du

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

        udu=2udu\int u\, du = - 2 \int u\, du

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

          udu=u22\int u\, du = \frac{u^{2}}{2}

        So, the result is: u2- u^{2}

      Now substitute uu back in:

      cos2(x2)- \cos^{2}{\left(\frac{x}{2} \right)}

  2. Now simplify:

    sin2(x2)\sin^{2}{\left(\frac{x}{2} \right)}

  3. Add the constant of integration:

    sin2(x2)+constant\sin^{2}{\left(\frac{x}{2} \right)}+ \mathrm{constant}


The answer is:

sin2(x2)+constant\sin^{2}{\left(\frac{x}{2} \right)}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                              
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sin(x2)cos(x2)dx=C+sin2(x2)\int \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}\, dx = C + \sin^{2}{\left(\frac{x}{2} \right)}
The graph
0.001.000.100.200.300.400.500.600.700.800.900.00.5
The answer [src]
   2     
sin (1/2)
sin2(12)\sin^{2}{\left(\frac{1}{2} \right)}
=
=
   2     
sin (1/2)
sin2(12)\sin^{2}{\left(\frac{1}{2} \right)}
sin(1/2)^2
Numerical answer [src]
0.22984884706593
0.22984884706593
The graph
Integral of sin(x/2)cos(x/2) dx

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