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Derivative of:
cos^2x/2
Identical expressions
cos^ two x/2
co sinus of e of squared x divide by 2
co sinus of e of to the power of two x divide by 2
cos2x/2
cos²x/2
cos to the power of 2x/2
cos^2x divide by 2
cos^2x/2dx

Solving integrals/ cos^2x/2

Integral of cos^2x/2 dx

The solution

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  1           
  /           
 |            
 |     2      
 |  cos (x)   
 |  ------- dx
 |     2      
 |            
/             
0

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

Integral(cos(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:

$\int \frac{\cos^{2}{\left(x \right)}}{2}\, dx = \frac{\int \cos^{2}{\left(x \right)}\, dx}{2}$
1. Rewrite the integrand:
  
  $\cos^{2}{\left(x \right)} = \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(2 x \right)}}{2}\, dx = \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
    1. Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{4}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
        
        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: $\frac{\sin{\left(2 x \right)}}{4}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{1}{2}\, dx = \frac{x}{2}$
  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                         
 | cos (x)          x   sin(2*x)
 | ------- dx = C + - + --------
 |    2             4      8    
 |                              
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   cos(1)*sin(1)
- + -------------
4         4

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

1   cos(1)*sin(1)
- + -------------
4         4

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

Упростить

Numerical answer [src]

0.36366217835321

0.36366217835321

The graph

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

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Identical expressions

Integral of cos^2x/2 dx

Limits of integration:

The graph:

Piecewise:

The solution