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Identical expressions
zero . five *sin^2x
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Solving integrals/ 0.5*sin^2x

Integral of 0.5*sin^2x dx

The solution

You have entered [src]

 pi           
 --           
 2            
  /           
 |            
 |     2      
 |  sin (x)   
 |  ------- dx
 |     2      
 |            
/             
pi            
--            
4

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

Integral(sin(x)^2/2, (x, pi/4, 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]

1   pi
- + --
8   16

$$\frac{1}{8} + \frac{\pi}{16}$$

1   pi
- + --
8   16

$$\frac{1}{8} + \frac{\pi}{16}$$

1/8 + pi/16

Numerical answer [src]

0.321349540849362

0.321349540849362

The graph

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

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Identical expressions

Integral of 0.5*sin^2x dx

Limits of integration:

The graph:

Piecewise:

The solution