Integral of (tgx+ctgx)² dx

The solution

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  1                      
  /                      
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 |                   2   
 |  (tan(x) + cot(x))  dx
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0

$$\int\limits_{0}^{1} \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right)^{2}\, dx$$

Simplify

Detail solution

Rewrite the integrand:

$\left(\tan{\left(x \right)} + \cot{\left(x \right)}\right)^{2} = \tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} \cot{\left(x \right)} + \cot^{2}{\left(x \right)}$
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:
  
  $\int 2 \tan{\left(x \right)} \cot{\left(x \right)}\, dx = 2 \int \tan{\left(x \right)} \cot{\left(x \right)}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $x$
  So, the result is: $2 x$
1. Don't know the steps in finding this integral.
  
  But the integral 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:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

$$\tan x-{{1}\over{\tan x}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$${\it \%a}$$

oo

$$\infty$$

Simplify

Numerical answer [src]

1.3793236779486e+19

1.3793236779486e+19

The graph

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

Other calculators

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Identical expressions

Similar expressions

Integral of (tgx+ctgx)² dx

Limits of integration:

The graph:

Piecewise:

The solution