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

Integral of cos(x/2)/2 dx

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∫01cos⁡(x2)2 dx\int\limits_{0}^{1} \frac{\cos{\left(\frac{x}{2} \right)}}{2}\, dx 
Integral(cos(x/2)/2, (x, 0, 1))
Detail solution

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

    ∫cos⁡(x2)2 dx=∫cos⁡(x2) dx2\int \frac{\cos{\left(\frac{x}{2} \right)}}{2}\, dx = \frac{\int \cos{\left(\frac{x}{2} \right)}\, dx}{2}

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

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

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

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

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

      Now substitute uu back in:

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

    So, the result is: sin⁡(x2)\sin{\left(\frac{x}{2} \right)}

  2. Now simplify:

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
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∫cos⁡(x2)2 dx=C+sin⁡(x2)\int \frac{\cos{\left(\frac{x}{2} \right)}}{2}\, dx = C + \sin{\left(\frac{x}{2} \right)}
Simplify
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The answer [src] 
sin(1/2)
sin⁡(12)\sin{\left(\frac{1}{2} \right)} 
=
= 
sin(1/2)
sin⁡(12)\sin{\left(\frac{1}{2} \right)}
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Numerical answer [src] 
0.479425538604203
0.479425538604203
The graph
Integral of cos(x/2)/2 dx

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

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