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Integral of (cos(1/x^2))*dx/x^3 dx

The solution

You have entered [src]

  1           
  /           
 |            
 |     /1 \   
 |  cos|--|   
 |     | 2|   
 |     \x /   
 |  ------- dx
 |      3     
 |     x      
 |            
/             
0

$$\int\limits_{0}^{1} \frac{\cos{\left(\frac{1}{x^{2}} \right)}}{x^{3}}\, dx$$

Integral(cos(1/(x^2))/x^3, (x, 0, 1))

Detail solution

Let $u = \frac{1}{x^{2}}$ .

Then let $du = - \frac{2 dx}{x^{3}}$ and substitute $- \frac{du}{2}$ :

$\int \left(- \frac{\cos{\left(u \right)}}{2}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \cos{\left(u \right)}\, du = - \frac{\int \cos{\left(u \right)}\, du}{2}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
  So, the result is: $- \frac{\sin{\left(u \right)}}{2}$
Now substitute $u$ back in:

$- \frac{\sin{\left(\frac{1}{x^{2}} \right)}}{2}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                        
 |                         
 |    /1 \             /1 \
 | cos|--|          sin|--|
 |    | 2|             | 2|
 |    \x /             \x /
 | ------- dx = C - -------
 |     3               2   
 |    x                    
 |                         
/

$$\int \frac{\cos{\left(\frac{1}{x^{2}} \right)}}{x^{3}}\, dx = C - \frac{\sin{\left(\frac{1}{x^{2}} \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   1   sin(1)  1   sin(1) 
<- - - ------, - - ------>
   2     2     2     2

$$\left\langle - \frac{1}{2} - \frac{\sin{\left(1 \right)}}{2}, \frac{1}{2} - \frac{\sin{\left(1 \right)}}{2}\right\rangle$$

   1   sin(1)  1   sin(1) 
<- - - ------, - - ------>
   2     2     2     2

$$\left\langle - \frac{1}{2} - \frac{\sin{\left(1 \right)}}{2}, \frac{1}{2} - \frac{\sin{\left(1 \right)}}{2}\right\rangle$$

AccumBounds(-1/2 - sin(1)/2, 1/2 - sin(1)/2)

Numerical answer [src]

2.03383349957075e+37

2.03383349957075e+37

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

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How to use it?

Identical expressions

Integral of (cos(1/x^2))*dx/x^3 dx

Limits of integration:

The graph:

Piecewise:

The solution