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Cos2x/(sin^2x*cos^2x)dx

Solving integrals/ Cos2x/(sin^2x*cos^2x)

Integral of Cos2x/(sin^2x*cos^2x) dx

The solution

You have entered [src]

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

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

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

Detail solution

Rewrite the integrand:

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

Упростить

Numerical answer [src]

1.3793236779486e+19

1.3793236779486e+19

The graph

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

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Identical expressions

Integral of Cos2x/(sin^2x*cos^2x) dx

Limits of integration:

The graph:

Piecewise:

The solution