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Solving integrals/ 2sin(x/2)*cos(x/2)

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

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The solution

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∫012sin⁡(x2)cos⁡(x2) dx\int\limits_{0}^{1} 2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}\, dx 
Integral((2*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=2sin⁡(x2)u = 2 \sin{\left(\frac{x}{2} \right)}.

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

      ∫u du\int u\, du

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

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

      Now substitute uu back in:

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

    Method #2

    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 4du4 du:

      ∫4u du\int 4 u\, du

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

        ∫u du=4∫u du\int u\, du = 4 \int u\, du

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

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

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

      Now substitute uu back in:

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

    Method #3

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

      Then let du=dx2du = \frac{dx}{2} and substitute 4du4 du:

      ∫4sin⁡(u)cos⁡(u) du\int 4 \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=4∫sin⁡(u)cos⁡(u) du\int \sin{\left(u \right)} \cos{\left(u \right)}\, du = 4 \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:

            ∫u du=−∫u du\int u\, du = - \int u\, du

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

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

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

          Now substitute uu back in:

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

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

      Now substitute uu back in:

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

    Method #4

    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 −4du- 4 du:

      ∫(−4u) du\int \left(- 4 u\right)\, du

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

        ∫u du=−4∫u du\int u\, du = - 4 \int u\, du

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

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

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

      Now substitute uu back in:

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

  2. Now simplify:

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

  3. Add the constant of integration:

    2sin⁡2(x2)+constant2 \sin^{2}{\left(\frac{x}{2} \right)}+ \mathrm{constant}

The answer is:

2sin⁡2(x2)+constant2 \sin^{2}{\left(\frac{x}{2} \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                  
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 |      /x\    /x\               2/x\
 | 2*sin|-|*cos|-| dx = C + 2*sin |-|
 |      \2/    \2/                \2/
 |                                   
/
∫2sin⁡(x2)cos⁡(x2) dx=C+2sin⁡2(x2)\int 2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}\, dx = C + 2 \sin^{2}{\left(\frac{x}{2} \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.01.0
Plot the graph f(x)
The answer [src] 
     2     
2*sin (1/2)
2sin⁡2(12)2 \sin^{2}{\left(\frac{1}{2} \right)} 
=
= 
     2     
2*sin (1/2)
2sin⁡2(12)2 \sin^{2}{\left(\frac{1}{2} \right)} 
2*sin(1/2)^2
Numerical answer [src] 
0.45969769413186
0.45969769413186

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

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