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Solving integrals/ 1/1-(cos(x))^2

Integral of 1/1-(cos(x))^2 dx

The solution

You have entered [src]

 pi                 
 --                 
 2                  
  /                 
 |                  
 |  /       2   \   
 |  \1 - cos (x)/ dx
 |                  
/                   
pi                  
--                  
4

$$\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{2}} \left(1 - \cos^{2}{\left(x \right)}\right)\, dx$$

Integral(1 - cos(x)^2, (x, pi/4, pi/2))

Detail solution

Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \cos^{2}{\left(x \right)}\right)\, dx = - \int \cos^{2}{\left(x \right)}\, dx$
  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 $\frac{du}{2}$ :
        
        $\int \frac{\cos{\left(u \right)}}{4}\, du$
        
        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: $- \frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}$
The result is: $\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                   
 |                                    
 | /       2   \          x   sin(2*x)
 | \1 - cos (x)/ dx = C + - - --------
 |                        2      4    
/

$$\int \left(1 - \cos^{2}{\left(x \right)}\right)\, dx = C + \frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   pi
- + --
4   8

$$\frac{1}{4} + \frac{\pi}{8}$$

1   pi
- + --
4   8

$$\frac{1}{4} + \frac{\pi}{8}$$

Упростить

Numerical answer [src]

0.642699081698724

0.642699081698724

The graph

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

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Integral of 1/1-(cos(x))^2 dx

Limits of integration:

The graph:

Piecewise:

The solution