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dx/1+cos^2(x)

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

The solution

You have entered [src]

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

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

Integral(1.0 - cos(x)^2, (x, pi, 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]

-0.25*pi

$$- 0.25 \pi$$

-0.25*pi

$$- 0.25 \pi$$

-0.25*pi

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 dx/1-cos^2(x) dx

Limits of integration:

The graph:

Piecewise:

The solution