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Solving integrals/ cos2x/sin^2(2x)

Integral of cos2x/sin^2(2x) dx

The solution

You have entered [src]

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

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

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

Detail solution

Let $u = 2 x$ .

Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :

$\int \frac{\cos{\left(u \right)}}{4 \sin^{2}{\left(u \right)}}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\cos{\left(u \right)}}{2 \sin^{2}{\left(u \right)}}\, du = \frac{\int \frac{\cos{\left(u \right)}}{\sin^{2}{\left(u \right)}}\, du}{2}$
  1. Let $u = \sin{\left(u \right)}$ .
    
    Then let $du = \cos{\left(u \right)} du$ and substitute $du$ :
    
    $\int \frac{1}{u^{2}}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
    Now substitute $u$ back in:
    
    $- \frac{1}{\sin{\left(u \right)}}$
  So, the result is: $- \frac{1}{2 \sin{\left(u \right)}}$
Now substitute $u$ back in:

$- \frac{1}{2 \sin{\left(2 x \right)}}$
Add the constant of integration:

$- \frac{1}{2 \sin{\left(2 x \right)}}+ \mathrm{constant}$

The answer is:

$- \frac{1}{2 \sin{\left(2 x \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

Упростить

Numerical answer [src]

3.44830919487149e+18

3.44830919487149e+18

The graph

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

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Identical expressions

Integral of cos2x/sin^2(2x) dx

Limits of integration:

The graph:

Piecewise:

The solution