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

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$$\int\limits_{0}^{1} \sin^{2}{\left(\frac{x}{2} \right)}\, dx$$

Integral(sin(x/2)^2, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\sin^{2}{\left(\frac{x}{2} \right)} = \frac{1}{2} - \frac{\cos{\left(x \right)}}{2}$
Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \frac{1}{2}\, dx = \frac{x}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\cos{\left(x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(x \right)}\, dx}{2}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  So, the result is: $- \frac{\sin{\left(x \right)}}{2}$
The result is: $\frac{x}{2} - \frac{\sin{\left(x \right)}}{2}$
Add the constant of integration:

$\frac{x}{2} - \frac{\sin{\left(x \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{x}{2} - \frac{\sin{\left(x \right)}}{2}+ \mathrm{constant}$

The graph

Plot the graph f(x)

The answer [src]

1/2 - cos(1/2)*sin(1/2)

$$- \sin{\left(\frac{1}{2} \right)} \cos{\left(\frac{1}{2} \right)} + \frac{1}{2}$$

1/2 - cos(1/2)*sin(1/2)

$$- \sin{\left(\frac{1}{2} \right)} \cos{\left(\frac{1}{2} \right)} + \frac{1}{2}$$

1/2 - cos(1/2)*sin(1/2)

Numerical answer [src]

0.0792645075960517

0.0792645075960517

The graph

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