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Integral of d{x}:
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Integral of csc^4(4x)
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Identical expressions
csc^ four (4x)
csc to the power of 4(4x)
csc to the power of four (4x)
csc4(4x)
csc44x
csc⁴(4x)
csc^44x
csc^4(4x)dx

Solving integrals/ csc^4(4x)

Integral of csc^4(4x) dx

The solution

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  1             
  /             
 |              
 |     4        
 |  csc (4*x) dx
 |              
/               
0

$$\int\limits_{0}^{1} \csc^{4}{\left(4 x \right)}\, dx$$

Integral(csc(4*x)^4, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\csc^{4}{\left(4 x \right)} = \left(\cot^{2}{\left(4 x \right)} + 1\right) \csc^{2}{\left(4 x \right)}$
Let $u = \cot{\left(4 x \right)}$ .

Then let $du = \left(- 4 \cot^{2}{\left(4 x \right)} - 4\right) dx$ and substitute $du$ :

$\int \left(- \frac{u^{2}}{4} - \frac{1}{4}\right)\, du$
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{u^{2}}{4}\right)\, du = - \frac{\int u^{2}\, du}{4}$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
    So, the result is: $- \frac{u^{3}}{12}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(- \frac{1}{4}\right)\, du = - \frac{u}{4}$
  The result is: $- \frac{u^{3}}{12} - \frac{u}{4}$
Now substitute $u$ back in:

$- \frac{\cot^{3}{\left(4 x \right)}}{12} - \frac{\cot{\left(4 x \right)}}{4}$
Now simplify:

$- \frac{\left(3 + \frac{1}{\tan^{2}{\left(4 x \right)}}\right) \cot{\left(4 x \right)}}{12}$
Add the constant of integration:

$- \frac{\left(3 + \frac{1}{\tan^{2}{\left(4 x \right)}}\right) \cot{\left(4 x \right)}}{12}+ \mathrm{constant}$

The answer is:

$- \frac{\left(3 + \frac{1}{\tan^{2}{\left(4 x \right)}}\right) \cot{\left(4 x \right)}}{12}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                       
 |                                  3     
 |    4               cot(4*x)   cot (4*x)
 | csc (4*x) dx = C - -------- - ---------
 |                       4           12   
/

$$-{{3\,\tan ^2\left(4\,x\right)+1}\over{12\,\tan ^3\left(4\,x\right) }}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$${\it \%a}$$

oo

$$\infty$$

Simplify

Numerical answer [src]

3.05246532205413e+54

3.05246532205413e+54

The graph

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

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Identical expressions

Integral of csc^4(4x) dx

Limits of integration:

The graph:

Piecewise:

The solution