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Integral of 1/cos^2x-sin^2x dx

The solution

You have entered [src]

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

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

Integral(1/(cos(x)^2) - sin(x)^2, (x, 0, 1))

Detail solution

Integrate term-by-term:
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: $- \frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
The result is: $- \frac{x}{2} + \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\sin{\left(2 x \right)}}{4}$
Now simplify:

$- \frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4} + \tan{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1   sin(1)   cos(1)*sin(1)
- - + ------ + -------------
  2   cos(1)         2

$$- \frac{1}{2} + \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{\sin{\left(1 \right)}}{\cos{\left(1 \right)}}$$

  1   sin(1)   cos(1)*sin(1)
- - + ------ + -------------
  2   cos(1)         2

$$- \frac{1}{2} + \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{\sin{\left(1 \right)}}{\cos{\left(1 \right)}}$$

-1/2 + sin(1)/cos(1) + cos(1)*sin(1)/2

Numerical answer [src]

1.28473208136132

1.28473208136132

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

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Integral of 1/cos^2x-sin^2x dx

Limits of integration:

The graph:

Piecewise:

The solution