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Integral of d{x}:
Integral of e^x/x^2
Integral of (1+x)/x
Integral of coshx
Integral of a^x*e^x
Graphing y =:
sin^2x/2
Identical expressions
sin^ two x/2
sinus of squared x divide by 2
sinus of to the power of two x divide by 2
sin2x/2
sin²x/2
sin to the power of 2x/2
sin^2x divide by 2
sin^2x/2dx
Similar expressions
(sin^2(x))/(2+3*cosx)

Integral of sin^2x/2 dx

The solution

You have entered [src]

  pi           
  --           
  2            
   /           
  |            
  |     2      
  |  sin (x)   
  |  ------- dx
  |     2      
  |            
 /             
3*pi           
----           
 2

$$\int\limits_{\frac{3 \pi}{2}}^{\frac{\pi}{2}} \frac{\sin^{2}{\left(x \right)}}{2}\, dx$$

Integral(sin(x)^2/2, (x, 3*pi/2, pi/2))

Detail solution

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

$\int \frac{\sin^{2}{\left(x \right)}}{2}\, dx = \frac{\int \sin^{2}{\left(x \right)}\, dx}{2}$
1. Rewrite the integrand:
  
  $\sin^{2}{\left(x \right)} = \frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2}$
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:
    
    $\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
    1. Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
        
        The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
        
        So, the result is: $\frac{\sin{\left(u \right)}}{2}$
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
    So, the result is: $- \frac{\sin{\left(2 x \right)}}{4}$
  The result is: $\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}$
So, the result is: $\frac{x}{4} - \frac{\sin{\left(2 x \right)}}{8}$
Add the constant of integration:

$\frac{x}{4} - \frac{\sin{\left(2 x \right)}}{8}+ \mathrm{constant}$

The answer is:

$\frac{x}{4} - \frac{\sin{\left(2 x \right)}}{8}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                             
 |                              
 |    2                         
 | sin (x)          sin(2*x)   x
 | ------- dx = C - -------- + -
 |    2                8       4
 |                              
/

$$\int \frac{\sin^{2}{\left(x \right)}}{2}\, dx = C + \frac{x}{4} - \frac{\sin{\left(2 x \right)}}{8}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-pi 
----
 4

$$- \frac{\pi}{4}$$

-pi 
----
 4

$$- \frac{\pi}{4}$$

-pi/4

Numerical answer [src]

-0.785398163397448

-0.785398163397448

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

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Identical expressions

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Integral of sin^2x/2 dx

Limits of integration:

The graph:

Piecewise:

The solution