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Similar expressions
sin((4/x)-5)dx/x^2

Integral of sin((4/x)+5)dx/x^2 dx

The solution

You have entered [src]

  1              
  /              
 |               
 |     /4    \   
 |  sin|- + 5|   
 |     \x    /   
 |  ---------- dx
 |       2       
 |      x        
 |               
/                
0

$$\int\limits_{0}^{1} \frac{\sin{\left(5 + \frac{4}{x} \right)}}{x^{2}}\, dx$$

Integral(sin(4/x + 5)/x^2, (x, 0, 1))

Detail solution

Let $u = 5 + \frac{4}{x}$ .

Then let $du = - \frac{4 dx}{x^{2}}$ and substitute $- \frac{du}{4}$ :

$\int \left(- \frac{\sin{\left(u \right)}}{4}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sin{\left(u \right)}\, du = - \frac{\int \sin{\left(u \right)}\, du}{4}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $\frac{\cos{\left(u \right)}}{4}$
Now substitute $u$ back in:

$\frac{\cos{\left(5 + \frac{4}{x} \right)}}{4}$
Now simplify:

$\frac{\cos{\left(5 + \frac{4}{x} \right)}}{4}$
Add the constant of integration:

$\frac{\cos{\left(5 + \frac{4}{x} \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{\cos{\left(5 + \frac{4}{x} \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                              
 |                               
 |    /4    \             /4    \
 | sin|- + 5|          cos|- + 5|
 |    \x    /             \x    /
 | ---------- dx = C + ----------
 |      2                  4     
 |     x                         
 |                               
/

$$\int \frac{\sin{\left(5 + \frac{4}{x} \right)}}{x^{2}}\, dx = C + \frac{\cos{\left(5 + \frac{4}{x} \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   1   cos(9)  1   cos(9) 
<- - + ------, - + ------>
   4     4     4     4

$$\left\langle - \frac{1}{4} + \frac{\cos{\left(9 \right)}}{4}, \frac{\cos{\left(9 \right)}}{4} + \frac{1}{4}\right\rangle$$

   1   cos(9)  1   cos(9) 
<- - + ------, - + ------>
   4     4     4     4

$$\left\langle - \frac{1}{4} + \frac{\cos{\left(9 \right)}}{4}, \frac{\cos{\left(9 \right)}}{4} + \frac{1}{4}\right\rangle$$

AccumBounds(-1/4 + cos(9)/4, 1/4 + cos(9)/4)

Numerical answer [src]

-2.77528115406785e+18

-2.77528115406785e+18

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

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Identical expressions

Similar expressions

Integral of sin((4/x)+5)dx/x^2 dx

Limits of integration:

The graph:

Piecewise:

The solution