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Identical expressions
tg^2x-ctg^2x
tg squared x minus ctg squared x
tg2x-ctg2x
tg²x-ctg²x
tg to the power of 2x-ctg to the power of 2x
tg^2x-ctg^2xdx
Similar expressions
tg^2x+ctg^2x

Integral of tg^2x-ctg^2x dx

The solution

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 pi                       
 --                       
 4                        
  /                       
 |                        
 |  /   2         2   \   
 |  \tan (x) - cot (x)/ dx
 |                        
/                         
pi                        
--                        
6

$$\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{4}} \left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right)\, dx$$

Integral(tan(x)^2 - cot(x)^2, (x, pi/6, pi/4))

Detail solution

Integrate term-by-term:
1. Rewrite the integrand:
  
  $\tan^{2}{\left(x \right)} = \sec^{2}{\left(x \right)} - 1$
2. Integrate term-by-term:
  1. $\int \sec^{2}{\left(x \right)}\, dx = \tan{\left(x \right)}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(-1\right)\, dx = - x$
  The result is: $- x + \tan{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \cot^{2}{\left(x \right)}\right)\, dx = - \int \cot^{2}{\left(x \right)}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $- x - \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
  So, the result is: $x + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
The result is: $\tan{\left(x \right)} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                            
 |                                             
 | /   2         2   \          cos(x)         
 | \tan (x) - cot (x)/ dx = C + ------ + tan(x)
 |                              sin(x)         
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

        ___
    4*\/ 3 
2 - -------
       3

$$2 - \frac{4 \sqrt{3}}{3}$$

        ___
    4*\/ 3 
2 - -------
       3

$$2 - \frac{4 \sqrt{3}}{3}$$

2 - 4*sqrt(3)/3

Numerical answer [src]

-0.309401076758503

-0.309401076758503

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

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Integral of tg^2x-ctg^2x dx

Limits of integration:

The graph:

Piecewise:

The solution