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Similar expressions
(cos(t^2)+1)/t^3

Integral of (cos(t^2)-1)/t^3 dt

The solution

You have entered [src]

  x               
  /               
 |                
 |     / 2\       
 |  cos\t / - 1   
 |  ----------- dt
 |        3       
 |       t        
 |                
/                 
0

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

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

Detail solution

Rewrite the integrand:

$\frac{\cos{\left(t^{2} \right)} - 1}{t^{3}} = \frac{\cos{\left(t^{2} \right)}}{t^{3}} - \frac{1}{t^{3}}$
Integrate term-by-term:
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $- \frac{\operatorname{Si}{\left(t^{2} \right)}}{2} - \frac{\cos{\left(t^{2} \right)}}{2 t^{2}}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{t^{3}}\right)\, dt = - \int \frac{1}{t^{3}}\, dt$
  1. The integral of $t^{n}$ is $\frac{t^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \frac{1}{t^{3}}\, dt = - \frac{1}{2 t^{2}}$
  So, the result is: $\frac{1}{2 t^{2}}$
The result is: $- \frac{\operatorname{Si}{\left(t^{2} \right)}}{2} - \frac{\cos{\left(t^{2} \right)}}{2 t^{2}} + \frac{1}{2 t^{2}}$
Now simplify:

$\frac{- t^{2} \operatorname{Si}{\left(t^{2} \right)} - \cos{\left(t^{2} \right)} + 1}{2 t^{2}}$
Add the constant of integration:

$\frac{- t^{2} \operatorname{Si}{\left(t^{2} \right)} - \cos{\left(t^{2} \right)} + 1}{2 t^{2}}+ \mathrm{constant}$

The answer is:

$\frac{- t^{2} \operatorname{Si}{\left(t^{2} \right)} - \cos{\left(t^{2} \right)} + 1}{2 t^{2}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                            
 |                                             
 |    / 2\                       / 2\      / 2\
 | cos\t / - 1           1     Si\t /   cos\t /
 | ----------- dt = C + ---- - ------ - -------
 |       3                 2     2           2 
 |      t               2*t               2*t  
 |                                             
/

$$\int \frac{\cos{\left(t^{2} \right)} - 1}{t^{3}}\, dt = C - \frac{\operatorname{Si}{\left(t^{2} \right)}}{2} - \frac{\cos{\left(t^{2} \right)}}{2 t^{2}} + \frac{1}{2 t^{2}}$$

Simplify

The answer [src]

         / 2\      / 2\
 1     Si\x /   cos\x /
---- - ------ - -------
   2     2           2 
2*x               2*x

$$- \frac{\operatorname{Si}{\left(x^{2} \right)}}{2} - \frac{\cos{\left(x^{2} \right)}}{2 x^{2}} + \frac{1}{2 x^{2}}$$

         / 2\      / 2\
 1     Si\x /   cos\x /
---- - ------ - -------
   2     2           2 
2*x               2*x

$$- \frac{\operatorname{Si}{\left(x^{2} \right)}}{2} - \frac{\cos{\left(x^{2} \right)}}{2 x^{2}} + \frac{1}{2 x^{2}}$$

1/(2*x^2) - Si(x^2)/2 - cos(x^2)/(2*x^2)

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

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Integral of (cos(t^2)-1)/t^3 dt

Limits of integration:

The graph:

Piecewise:

The solution