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Integral of sin(1/x)*(dx/x^2) dx

The solution

You have entered [src]

  1          
  /          
 |           
 |     /1\   
 |  sin|-|   
 |     \x/   
 |  ------ dx
 |     2     
 |    x      
 |           
/            
0

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

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

Detail solution

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

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

<-1 + cos(1), 1 + cos(1)>

$$\left\langle -1 + \cos{\left(1 \right)}, \cos{\left(1 \right)} + 1\right\rangle$$

<-1 + cos(1), 1 + cos(1)>

$$\left\langle -1 + \cos{\left(1 \right)}, \cos{\left(1 \right)} + 1\right\rangle$$

AccumBounds(-1 + cos(1), 1 + cos(1))

Numerical answer [src]

-1.74295524279167e+18

-1.74295524279167e+18

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

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Identical expressions

Integral of sin(1/x)*(dx/x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution