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Solving integrals/ (sin(sqrt(x)+4))/(sqrt(x))

Integral of (sin(sqrt(x)+4))/(sqrt(x)) dx

The solution

You have entered [src]

  1                  
  /                  
 |                   
 |     /  ___    \   
 |  sin\\/ x  + 4/   
 |  -------------- dx
 |        ___        
 |      \/ x         
 |                   
/                    
0

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

Integral(sin(sqrt(x) + 4)/sqrt(x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sqrt{x} + 4$ .
  
  Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$ :
  
  $\int 2 \sin{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = 2 \int \sin{\left(u \right)}\, du$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- 2 \cos{\left(u \right)}$
  Now substitute $u$ back in:
  
  $- 2 \cos{\left(\sqrt{x} + 4 \right)}$
Method #2
1. Let $u = \sqrt{x}$ .
  
  Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$ :
  
  $\int 2 \sin{\left(u + 4 \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u + 4 \right)}\, du = 2 \int \sin{\left(u + 4 \right)}\, du$
    1. Let $u = u + 4$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \sin{\left(u \right)}\, du$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- \cos{\left(u + 4 \right)}$
    So, the result is: $- 2 \cos{\left(u + 4 \right)}$
  Now substitute $u$ back in:
  
  $- 2 \cos{\left(\sqrt{x} + 4 \right)}$
Now simplify:

$- 2 \cos{\left(\sqrt{x} + 4 \right)}$
Add the constant of integration:

$- 2 \cos{\left(\sqrt{x} + 4 \right)}+ \mathrm{constant}$

The answer is:

$- 2 \cos{\left(\sqrt{x} + 4 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-2*cos(5) + 2*cos(4)

$$2 \cos{\left(4 \right)} - 2 \cos{\left(5 \right)}$$

-2*cos(5) + 2*cos(4)

$$2 \cos{\left(4 \right)} - 2 \cos{\left(5 \right)}$$

-2*cos(5) + 2*cos(4)

Numerical answer [src]

-1.87461161225213

-1.87461161225213

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

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Identical expressions

Similar expressions

Integral of (sin(sqrt(x)+4))/(sqrt(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2