Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of |x-1|
Integral of (x+2)dx
Integral of (3x-2)^2
Integral of x(x^2+3)
Identical expressions
cot(x)csc^ two (x)
cotangent of (x)csc squared (x)
cotangent of (x)csc to the power of two (x)
cot(x)csc2(x)
cotxcsc2x
cot(x)csc²(x)
cot(x)csc to the power of 2(x)
cotxcsc^2x
cot(x)csc^2(x)dx

Integral of cot(x)csc^2(x) dx

The solution

You have entered [src]

  pi                  
  --                  
  2                   
   /                  
  |                   
  |            2      
  |  cot(x)*csc (x) dx
  |                   
 /                    
3*pi                  
----                  
 2

$$\int\limits_{\frac{3 \pi}{2}}^{\frac{\pi}{2}} \cot{\left(x \right)} \csc^{2}{\left(x \right)}\, dx$$

Integral(cot(x)*csc(x)^2, (x, 3*pi/2, pi/2))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \cot{\left(x \right)}$ .
  
  Then let $du = \left(- \cot^{2}{\left(x \right)} - 1\right) dx$ and substitute $- du$ :
  
  $\int \left(- u\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int u\, du = - \int u\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
    So, the result is: $- \frac{u^{2}}{2}$
  Now substitute $u$ back in:
  
  $- \frac{\cot^{2}{\left(x \right)}}{2}$
Method #2
1. Let $u = \csc{\left(x \right)}$ .
  
  Then let $du = - \cot{\left(x \right)} \csc{\left(x \right)} dx$ and substitute $- du$ :
  
  $\int \left(- u\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int u\, du = - \int u\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
    So, the result is: $- \frac{u^{2}}{2}$
  Now substitute $u$ back in:
  
  $- \frac{\csc^{2}{\left(x \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                               
 |                            2   
 |           2             cot (x)
 | cot(x)*csc (x) dx = C - -------
 |                            2   
/

$$\int \cot{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = C - \frac{\cot^{2}{\left(x \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

nan

$$\text{NaN}$$

nan

$$\text{NaN}$$

nan

Numerical answer [src]

2.0990714654111e+46

2.0990714654111e+46

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

Other calculators

How to use it?

Identical expressions

Integral of cot(x)csc^2(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2