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Identical expressions
(x+ one)/(x+ six)^(one / two)
(x plus 1) divide by (x plus 6) to the power of (1 divide by 2)
(x plus one) divide by (x plus six) to the power of (one divide by two)
(x+1)/(x+6)(1/2)
x+1/x+61/2
x+1/x+6^1/2
(x+1) divide by (x+6)^(1 divide by 2)
(x+1)/(x+6)^(1/2)dx
Similar expressions
(x-1)/(x+6)^(1/2)
(x+1)/(x-6)^(1/2)

Integral of (x+1)/(x+6)^(1/2) dx

The solution

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 -2             
  /             
 |              
 |    x + 1     
 |  --------- dx
 |    _______   
 |  \/ x + 6    
 |              
/               
-5

$$\int\limits_{-5}^{-2} \frac{x + 1}{\sqrt{x + 6}}\, dx$$

Integral((x + 1)/sqrt(x + 6), (x, -5, -2))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sqrt{x + 6}$ .
  
  Then let $du = \frac{dx}{2 \sqrt{x + 6}}$ and substitute $du$ :
  
  $\int \left(2 u^{2} - 10\right)\, du$
  1. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 u^{2}\, du = 2 \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $\frac{2 u^{3}}{3}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int \left(-10\right)\, du = - 10 u$
    The result is: $\frac{2 u^{3}}{3} - 10 u$
  Now substitute $u$ back in:
  
  $\frac{2 \left(x + 6\right)^{\frac{3}{2}}}{3} - 10 \sqrt{x + 6}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x + 1}{\sqrt{x + 6}} = \frac{x}{\sqrt{x + 6}} + \frac{1}{\sqrt{x + 6}}$
2. Integrate term-by-term:
  1. Let $u = \frac{1}{\sqrt{x + 6}}$ .
    
    Then let $du = - \frac{dx}{2 \left(x + 6\right)^{\frac{3}{2}}}$ and substitute $du$ :
    
    $\int \left(- 2 \left(-6 + \frac{1}{u^{2}}\right)^{2} + 72 - \frac{12}{u^{2}}\right)\, du$
    1. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 2 \left(-6 + \frac{1}{u^{2}}\right)^{2}\right)\, du = - 2 \int \left(-6 + \frac{1}{u^{2}}\right)^{2}\, du$
      
      There are multiple ways to do this integral.
      
      Method #1
      
      Rewrite the integrand:
      
      $\left(-6 + \frac{1}{u^{2}}\right)^{2} = 36 - \frac{12}{u^{2}} + \frac{1}{u^{4}}$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 36\, du = 36 u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{12}{u^{2}}\right)\, du = - 12 \int \frac{1}{u^{2}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
      
      So, the result is: $\frac{12}{u}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{4}}\, du = - \frac{1}{3 u^{3}}$
      
      The result is: $36 u + \frac{12}{u} - \frac{1}{3 u^{3}}$
      
      Method #2
      
      Rewrite the integrand:
      
      $\left(-6 + \frac{1}{u^{2}}\right)^{2} = \frac{36 u^{4} - 12 u^{2} + 1}{u^{4}}$
      
      Rewrite the integrand:
      
      $\frac{36 u^{4} - 12 u^{2} + 1}{u^{4}} = 36 - \frac{12}{u^{2}} + \frac{1}{u^{4}}$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 36\, du = 36 u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{12}{u^{2}}\right)\, du = - 12 \int \frac{1}{u^{2}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
      
      So, the result is: $\frac{12}{u}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{4}}\, du = - \frac{1}{3 u^{3}}$
      
      The result is: $36 u + \frac{12}{u} - \frac{1}{3 u^{3}}$
      
      So, the result is: $- 72 u - \frac{24}{u} + \frac{2}{3 u^{3}}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 72\, du = 72 u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{12}{u^{2}}\right)\, du = - 12 \int \frac{1}{u^{2}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
      
      So, the result is: $\frac{12}{u}$
      
      The result is: $- \frac{12}{u} + \frac{2}{3 u^{3}}$
    Now substitute $u$ back in:
    
    $\frac{2 \left(x + 6\right)^{\frac{3}{2}}}{3} - 12 \sqrt{x + 6}$
  1. Let $u = x + 6$ .
    
    Then let $du = dx$ and substitute $du$ :
    
    $\int \frac{1}{\sqrt{u}}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{\sqrt{u}}\, du = 2 \sqrt{u}$
    Now substitute $u$ back in:
    
    $2 \sqrt{x + 6}$
  The result is: $\frac{2 \left(x + 6\right)^{\frac{3}{2}}}{3} - 10 \sqrt{x + 6}$
Now simplify:

$\frac{2 \left(x - 9\right) \sqrt{x + 6}}{3}$
Add the constant of integration:

$\frac{2 \left(x - 9\right) \sqrt{x + 6}}{3}+ \mathrm{constant}$

The answer is:

$\frac{2 \left(x - 9\right) \sqrt{x + 6}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                              
 |                                            3/2
 |   x + 1                 _______   2*(x + 6)   
 | --------- dx = C - 10*\/ x + 6  + ------------
 |   _______                              3      
 | \/ x + 6                                      
 |                                               
/

$$\int \frac{x + 1}{\sqrt{x + 6}}\, dx = C + \frac{2 \left(x + 6\right)^{\frac{3}{2}}}{3} - 10 \sqrt{x + 6}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-16/3

$$- \frac{16}{3}$$

-16/3

$$- \frac{16}{3}$$

-16/3

Numerical answer [src]

-5.33333333333333

-5.33333333333333

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

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Identical expressions

Similar expressions

Integral of (x+1)/(x+6)^(1/2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #1

Method #2