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Identical expressions
dx/(sqrtx+ five)
dx divide by ( square root of x plus 5)
dx divide by ( square root of x plus five)
dx/(√x+5)
dx/sqrtx+5
dx divide by (sqrtx+5)
Similar expressions
dx/(sqrtx-5)

Integral of dx/(sqrtx+5) dx

The solution

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  1             
  /             
 |              
 |      1       
 |  --------- dx
 |    ___       
 |  \/ x  + 5   
 |              
/               
0

$$\int\limits_{0}^{1} \frac{1}{\sqrt{x} + 5}\, dx$$

Integral(1/(sqrt(x) + 5), (x, 0, 1))

Detail solution

Let $u = \sqrt{x}$ .

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

$\int \frac{2 u}{u + 5}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{u}{u + 5}\, du = 2 \int \frac{u}{u + 5}\, du$
  1. Rewrite the integrand:
    
    $\frac{u}{u + 5} = 1 - \frac{5}{u + 5}$
  2. Integrate term-by-term:
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{5}{u + 5}\right)\, du = - 5 \int \frac{1}{u + 5}\, du$
      1. Let $u = u + 5$ .
        
        Then let $du = du$ and substitute $du$ :
        
        $\int \frac{1}{u}\, du$
        
        The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
        
        Now substitute $u$ back in:
        
        $\log{\left(u + 5 \right)}$
      So, the result is: $- 5 \log{\left(u + 5 \right)}$
    The result is: $u - 5 \log{\left(u + 5 \right)}$
  So, the result is: $2 u - 10 \log{\left(u + 5 \right)}$
Now substitute $u$ back in:

$2 \sqrt{x} - 10 \log{\left(\sqrt{x} + 5 \right)}$
Add the constant of integration:

$2 \sqrt{x} - 10 \log{\left(\sqrt{x} + 5 \right)}+ \mathrm{constant}$

The answer is:

$2 \sqrt{x} - 10 \log{\left(\sqrt{x} + 5 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                              
 |                                               
 |     1                    /      ___\       ___
 | --------- dx = C - 10*log\5 + \/ x / + 2*\/ x 
 |   ___                                         
 | \/ x  + 5                                     
 |                                               
/

$$\int \frac{1}{\sqrt{x} + 5}\, dx = C + 2 \sqrt{x} - 10 \log{\left(\sqrt{x} + 5 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

2 - 10*log(6) + 10*log(5)

$$- 10 \log{\left(6 \right)} + 2 + 10 \log{\left(5 \right)}$$

2 - 10*log(6) + 10*log(5)

$$- 10 \log{\left(6 \right)} + 2 + 10 \log{\left(5 \right)}$$

2 - 10*log(6) + 10*log(5)

Numerical answer [src]

0.176784432060454

0.176784432060454

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

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Integral of dx/(sqrtx+5) dx

Limits of integration:

The graph:

Piecewise:

The solution