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Similar expressions
dx/1+sin^2x

Integral of dx/1-sin^2x dx

The solution

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  pi                   
  --                   
  4                    
   /                   
  |                    
  |  /         2   \   
  |  \1.0 - sin (x)/ dx
  |                    
 /                     
-pi                    
----                   
 4

$$\int\limits_{- \frac{\pi}{4}}^{\frac{\pi}{4}} \left(1.0 - \sin^{2}{\left(x \right)}\right)\, dx$$

Integral(1.0 - sin(x)^2, (x, -pi/4, pi/4))

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:
  
  $\int \left(- \sin^{2}{\left(x \right)}\right)\, dx = - \int \sin^{2}{\left(x \right)}\, dx$
  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 $\frac{du}{2}$ :
        
        $\int \frac{\cos{\left(u \right)}}{2}\, du$
        
        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: $- \frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}$
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 - sin (x)/ dx = C + -------- + 0.5*x
 |                             4            
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/2 + 0.25*pi

$$\frac{1}{2} + 0.25 \pi$$

1/2 + 0.25*pi

$$\frac{1}{2} + 0.25 \pi$$

1/2 + 0.25*pi

Numerical answer [src]

1.28539816339745

1.28539816339745

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

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Integral of dx/1-sin^2x dx

Limits of integration:

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The solution