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Identical expressions
sqrt(sinx)/x^ two (x+ four)
square root of ( sinus of x) divide by x squared (x plus 4)
square root of ( sinus of x) divide by x to the power of two (x plus four)
√(sinx)/x^2(x+4)
sqrt(sinx)/x2(x+4)
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sqrt(sinx)/x^2(x+4)dx
Similar expressions
sqrt(sinx)/x^2(x-4)

Integral of sqrt(sinx)/x^2(x+4) dx

The solution

You have entered [src]

 oo                      
  /                      
 |                       
 |    ________           
 |  \/ sin(x) *(x + 4)   
 |  ------------------ dx
 |           2           
 |          x            
 |                       
/                        
2

$$\int\limits_{2}^{\infty} \frac{\left(x + 4\right) \sqrt{\sin{\left(x \right)}}}{x^{2}}\, dx$$

Integral(sqrt(sin(x))*(x + 4)/(x^2), (x, 2, oo))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{\left(x + 4\right) \sqrt{\sin{\left(x \right)}}}{x^{2}} = \frac{x \sqrt{\sin{\left(x \right)}} + 4 \sqrt{\sin{\left(x \right)}}}{x^{2}}$
2. Rewrite the integrand:
  
  $\frac{x \sqrt{\sin{\left(x \right)}} + 4 \sqrt{\sin{\left(x \right)}}}{x^{2}} = \frac{\sqrt{\sin{\left(x \right)}}}{x} + \frac{4 \sqrt{\sin{\left(x \right)}}}{x^{2}}$
3. Integrate term-by-term:
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\int \frac{\sqrt{\sin{\left(x \right)}}}{x}\, dx$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{4 \sqrt{\sin{\left(x \right)}}}{x^{2}}\, dx = 4 \int \frac{\sqrt{\sin{\left(x \right)}}}{x^{2}}\, dx$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\int \frac{\sqrt{\sin{\left(x \right)}}}{x^{2}}\, dx$
    So, the result is: $4 \int \frac{\sqrt{\sin{\left(x \right)}}}{x^{2}}\, dx$
  The result is: $4 \int \frac{\sqrt{\sin{\left(x \right)}}}{x^{2}}\, dx + \int \frac{\sqrt{\sin{\left(x \right)}}}{x}\, dx$
Method #2
1. Rewrite the integrand:
  
  $\frac{\left(x + 4\right) \sqrt{\sin{\left(x \right)}}}{x^{2}} = \frac{\sqrt{\sin{\left(x \right)}}}{x} + \frac{4 \sqrt{\sin{\left(x \right)}}}{x^{2}}$
2. Integrate term-by-term:
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\int \frac{\sqrt{\sin{\left(x \right)}}}{x}\, dx$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{4 \sqrt{\sin{\left(x \right)}}}{x^{2}}\, dx = 4 \int \frac{\sqrt{\sin{\left(x \right)}}}{x^{2}}\, dx$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\int \frac{\sqrt{\sin{\left(x \right)}}}{x^{2}}\, dx$
    So, the result is: $4 \int \frac{\sqrt{\sin{\left(x \right)}}}{x^{2}}\, dx$
  The result is: $4 \int \frac{\sqrt{\sin{\left(x \right)}}}{x^{2}}\, dx + \int \frac{\sqrt{\sin{\left(x \right)}}}{x}\, dx$
Add the constant of integration:

$4 \int \frac{\sqrt{\sin{\left(x \right)}}}{x^{2}}\, dx + \int \frac{\sqrt{\sin{\left(x \right)}}}{x}\, dx+ \mathrm{constant}$

The answer is:

$4 \int \frac{\sqrt{\sin{\left(x \right)}}}{x^{2}}\, dx + \int \frac{\sqrt{\sin{\left(x \right)}}}{x}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                /                  /             
 |                                |                  |              
 |   ________                     |   ________       |   ________   
 | \/ sin(x) *(x + 4)             | \/ sin(x)        | \/ sin(x)    
 | ------------------ dx = C + 4* | ---------- dx +  | ---------- dx
 |          2                     |      2           |     x        
 |         x                      |     x            |              
 |                                |                 /               
/                                /

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

Simplify

The answer [src]

 oo                      
  /                      
 |                       
 |    ________           
 |  \/ sin(x) *(4 + x)   
 |  ------------------ dx
 |           2           
 |          x            
 |                       
/                        
2

$$\int\limits_{2}^{\infty} \frac{\left(x + 4\right) \sqrt{\sin{\left(x \right)}}}{x^{2}}\, dx$$

 oo                      
  /                      
 |                       
 |    ________           
 |  \/ sin(x) *(4 + x)   
 |  ------------------ dx
 |           2           
 |          x            
 |                       
/                        
2

$$\int\limits_{2}^{\infty} \frac{\left(x + 4\right) \sqrt{\sin{\left(x \right)}}}{x^{2}}\, dx$$

Упростить

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of sqrt(sinx)/x^2(x+4) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2