Integral of dx/1-cos^2x dx

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 pi                   
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 2                    
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 |  /         2   \   
 |  \1.0 - cos (x)/ dx
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40

$$\int\limits_{\frac{\pi}{40}}^{\frac{\pi}{2}} \left(1.0 - \cos^{2}{\left(x \right)}\right)\, dx$$

Integral(1.0 - cos(x)^2, (x, pi/40, pi/2))

Detail solution

Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1.0\, dx = 1.0 x$
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 $\frac{du}{2}$ :
        
        $\int \frac{\cos{\left(u \right)}}{2}\, du$
        
        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: $\frac{\sin{\left(2 x \right)}}{4}$
    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: $0.5 x - \frac{\sin{\left(2 x \right)}}{4}$
Add the constant of integration:

$0.5 x - \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}$

The answer is:

$0.5 x - \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                         
 |                                          
 | /         2   \          sin(2*x)        
 | \1.0 - cos (x)/ dx = C - -------- + 0.5*x
 |                             4            
/

$$\int \left(1.0 - \cos^{2}{\left(x \right)}\right)\, dx = C + 0.5 x - \frac{\sin{\left(2 x \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

            /     ___________      ___________        ___________            \ /     ___________                      ___________      ___________\
            |    /       ___      /       ___        /       ___  /      ___\| |    /       ___  /        ___\       /       ___      /       ___ |
            |   /  1   \/ 2      /  5   \/ 5        /  1   \/ 2   |1   \/ 5 || |   /  1   \/ 2   |  1   \/ 5 |      /  1   \/ 2      /  5   \/ 5  |
            |  /   - - ----- *  /   - + -----  +   /   - + ----- *|- - -----||*|  /   - - ----- *|- - + -----| +   /   - + ----- *  /   - + ----- |
            \\/    2     4    \/    8     8      \/    2     4    \4     4  // \\/    2     4    \  4     4  /   \/    2     4    \/    8     8   /
0.2375*pi + ---------------------------------------------------------------------------------------------------------------------------------------
                                                                               2

$$\frac{\left(\left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right) \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right) \left(\left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right) \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)}{2} + 0.2375 \pi$$

=

            /     ___________      ___________        ___________            \ /     ___________                      ___________      ___________\
            |    /       ___      /       ___        /       ___  /      ___\| |    /       ___  /        ___\       /       ___      /       ___ |
            |   /  1   \/ 2      /  5   \/ 5        /  1   \/ 2   |1   \/ 5 || |   /  1   \/ 2   |  1   \/ 5 |      /  1   \/ 2      /  5   \/ 5  |
            |  /   - - ----- *  /   - + -----  +   /   - + ----- *|- - -----||*|  /   - - ----- *|- - + -----| +   /   - + ----- *  /   - + ----- |
            \\/    2     4    \/    8     8      \/    2     4    \4     4  // \\/    2     4    \  4     4  /   \/    2     4    \/    8     8   /
0.2375*pi + ---------------------------------------------------------------------------------------------------------------------------------------
                                                                               2

$$\frac{\left(\left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right) \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right) \left(\left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right) \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)}{2} + 0.2375 \pi$$

0.2375*pi + (sqrt(1/2 - sqrt(2)/4)*sqrt(5/8 + sqrt(5)/8) + sqrt(1/2 + sqrt(2)/4)*(1/4 - sqrt(5)/4))*(sqrt(1/2 - sqrt(2)/4)*(-1/4 + sqrt(5)/4) + sqrt(1/2 + sqrt(2)/4)*sqrt(5/8 + sqrt(5)/8))/2

Numerical answer [src]

0.785236871487634

0.785236871487634

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Integral of dx/1-cos^2x dx

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