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4*sin(x/2)^(2)dx

Integral of 4*sin(x/2)^(2) dx

The solution

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  1             
  /             
 |              
 |       2/x\   
 |  4*sin |-| dx
 |        \2/   
 |              
/               
0

$$\int\limits_{0}^{1} 4 \sin^{2}{\left(\frac{x}{2} \right)}\, dx$$

Integral(4*sin(x/2)^2, (x, 0, 1))

Detail solution

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

$\int 4 \sin^{2}{\left(\frac{x}{2} \right)}\, dx = 4 \int \sin^{2}{\left(\frac{x}{2} \right)}\, dx$
1. Rewrite the integrand:
  
  $\sin^{2}{\left(\frac{x}{2} \right)} = \frac{1}{2} - \frac{\cos{\left(x \right)}}{2}$
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: $\frac{x}{2} - \frac{\sin{\left(x \right)}}{2}$
So, the result is: $2 x - 2 \sin{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                 
 |                                  
 |      2/x\                        
 | 4*sin |-| dx = C - 2*sin(x) + 2*x
 |       \2/                        
 |                                  
/

$$\int 4 \sin^{2}{\left(\frac{x}{2} \right)}\, dx = C + 2 x - 2 \sin{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

2 - 4*cos(1/2)*sin(1/2)

$$- 4 \sin{\left(\frac{1}{2} \right)} \cos{\left(\frac{1}{2} \right)} + 2$$

2 - 4*cos(1/2)*sin(1/2)

$$- 4 \sin{\left(\frac{1}{2} \right)} \cos{\left(\frac{1}{2} \right)} + 2$$

2 - 4*cos(1/2)*sin(1/2)

Numerical answer [src]

0.317058030384207

0.317058030384207

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

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Identical expressions

Integral of 4*sin(x/2)^(2) dx

Limits of integration:

The graph:

Piecewise:

The solution