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

The solution

You have entered [src]

 99              
  /              
 |               
 |         /1\   
 |  x - sin|-|   
 |         \x/   
 |  ---------- dx
 |       2       
 |      x        
 |               
/                
15

$$\int\limits_{15}^{99} \frac{x - \sin{\left(\frac{1}{x} \right)}}{x^{2}}\, dx$$

Integral((x - sin(1/x))/x^2, (x, 15, 99))

Detail solution

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

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-cos(1/99) - log(15) + cos(1/15) + log(99)

$$- \log{\left(15 \right)} - \cos{\left(\frac{1}{99} \right)} + \cos{\left(\frac{1}{15} \right)} + \log{\left(99 \right)}$$

-cos(1/99) - log(15) + cos(1/15) + log(99)

$$- \log{\left(15 \right)} - \cos{\left(\frac{1}{99} \right)} + \cos{\left(\frac{1}{15} \right)} + \log{\left(99 \right)}$$

-cos(1/99) - log(15) + cos(1/15) + log(99)

Numerical answer [src]

1.88489926450228

1.88489926450228

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution