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Identical expressions
dx/(one -sqrt(x+ fifteen))
dx divide by (1 minus square root of (x plus 15))
dx divide by (one minus square root of (x plus fifteen))
dx/(1-√(x+15))
dx/1-sqrtx+15
dx divide by (1-sqrt(x+15))
Similar expressions
dx/(1+sqrt(x+15))
dx/(1-sqrt(x-15))

Integral of dx/(1-sqrt(x+15)) dx

The solution

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 10                  
  /                  
 |                   
 |        1          
 |  -------------- dx
 |        ________   
 |  1 - \/ x + 15    
 |                   
/                    
1

$$\int\limits_{1}^{10} \frac{1}{1 - \sqrt{x + 15}}\, dx$$

Integral(1/(1 - sqrt(x + 15)), (x, 1, 10))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sqrt{x + 15}$ .
  
  Then let $du = \frac{dx}{2 \sqrt{x + 15}}$ and substitute $- 2 du$ :
  
  $\int \left(- \frac{2 u}{u - 1}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{u}{u - 1}\, du = - 2 \int \frac{u}{u - 1}\, du$
    1. Rewrite the integrand:
      
      $\frac{u}{u - 1} = 1 + \frac{1}{u - 1}$
    2. Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      Let $u = u - 1$ .
      
      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:
      
      $\log{\left(u - 1 \right)}$
      
      The result is: $u + \log{\left(u - 1 \right)}$
    So, the result is: $- 2 u - 2 \log{\left(u - 1 \right)}$
  Now substitute $u$ back in:
  
  $- 2 \sqrt{x + 15} - 2 \log{\left(\sqrt{x + 15} - 1 \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{1}{1 - \sqrt{x + 15}} = - \frac{1}{\sqrt{x + 15} - 1}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{\sqrt{x + 15} - 1}\right)\, dx = - \int \frac{1}{\sqrt{x + 15} - 1}\, dx$
  1. Let $u = \sqrt{x + 15}$ .
    
    Then let $du = \frac{dx}{2 \sqrt{x + 15}}$ and substitute $2 du$ :
    
    $\int \frac{2 u}{u - 1}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{u}{u - 1}\, du = 2 \int \frac{u}{u - 1}\, du$
      
      Rewrite the integrand:
      
      $\frac{u}{u - 1} = 1 + \frac{1}{u - 1}$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      Let $u = u - 1$ .
      
      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:
      
      $\log{\left(u - 1 \right)}$
      
      The result is: $u + \log{\left(u - 1 \right)}$
      
      So, the result is: $2 u + 2 \log{\left(u - 1 \right)}$
    Now substitute $u$ back in:
    
    $2 \sqrt{x + 15} + 2 \log{\left(\sqrt{x + 15} - 1 \right)}$
  So, the result is: $- 2 \sqrt{x + 15} - 2 \log{\left(\sqrt{x + 15} - 1 \right)}$
Now simplify:

$- 2 \sqrt{x + 15} - 2 \log{\left(\sqrt{x + 15} - 1 \right)}$
Add the constant of integration:

$- 2 \sqrt{x + 15} - 2 \log{\left(\sqrt{x + 15} - 1 \right)}+ \mathrm{constant}$

The answer is:

$- 2 \sqrt{x + 15} - 2 \log{\left(\sqrt{x + 15} - 1 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                             
 |                                                              
 |       1                     ________        /       ________\
 | -------------- dx = C - 2*\/ x + 15  - 2*log\-1 + \/ x + 15 /
 |       ________                                               
 | 1 - \/ x + 15                                                
 |                                                              
/

$$\int \frac{1}{1 - \sqrt{x + 15}}\, dx = C - 2 \sqrt{x + 15} - 2 \log{\left(\sqrt{x + 15} - 1 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-2 - 2*log(4) + 2*log(3)

$$- 2 \log{\left(4 \right)} - 2 + 2 \log{\left(3 \right)}$$

-2 - 2*log(4) + 2*log(3)

$$- 2 \log{\left(4 \right)} - 2 + 2 \log{\left(3 \right)}$$

-2 - 2*log(4) + 2*log(3)

Numerical answer [src]

-2.57536414490356

-2.57536414490356

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

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Identical expressions

Similar expressions

Integral of dx/(1-sqrt(x+15)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2