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Identical expressions
2x/(✓x²+✓ eight)dx
2x divide by (✓x² plus ✓8)dx
2x divide by (✓x² plus ✓ eight)dx
2x/✓x²+✓8dx
2x divide by (✓x²+✓8)dx
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2x/(✓x²-✓8)dx

Integral of 2x/(✓x²+✓8)dx dx

The solution

You have entered [src]

  4                  
  /                  
 |                   
 |       2*x         
 |  -------------- dx
 |       2           
 |    ___      ___   
 |  \/ x   + \/ 8    
 |                   
/                    
2

$$\int\limits_{2}^{4} \frac{2 x}{\left(\sqrt{x}\right)^{2} + \sqrt{8}}\, dx$$

Integral((2*x)/((sqrt(x))^2 + sqrt(8)), (x, 2, 4))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sqrt{x}$ .
  
  Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $4 du$ :
  
  $\int \frac{4 u^{3}}{u^{2} + 2 \sqrt{2}}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{u^{3}}{u^{2} + 2 \sqrt{2}}\, du = 4 \int \frac{u^{3}}{u^{2} + 2 \sqrt{2}}\, du$
    1. Let $u = u^{2}$ .
      
      Then let $du = 2 u du$ and substitute $du$ :
      
      $\int \frac{u}{2 u + 4 \sqrt{2}}\, du$
      
      Rewrite the integrand:
      
      $\frac{u}{2 u + 4 \sqrt{2}} = \frac{1}{2} - \frac{\sqrt{2}}{u + 2 \sqrt{2}}$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, du = \frac{u}{2}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{\sqrt{2}}{u + 2 \sqrt{2}}\right)\, du = - \sqrt{2} \int \frac{1}{u + 2 \sqrt{2}}\, du$
      
      Let $u = u + 2 \sqrt{2}$ .
      
      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:
      
      $\log{\left(u + 2 \sqrt{2} \right)}$
      
      So, the result is: $- \sqrt{2} \log{\left(u + 2 \sqrt{2} \right)}$
      
      The result is: $\frac{u}{2} - \sqrt{2} \log{\left(u + 2 \sqrt{2} \right)}$
      
      Now substitute $u$ back in:
      
      $\frac{u^{2}}{2} - \sqrt{2} \log{\left(u^{2} + 2 \sqrt{2} \right)}$
    So, the result is: $2 u^{2} - 4 \sqrt{2} \log{\left(u^{2} + 2 \sqrt{2} \right)}$
  Now substitute $u$ back in:
  
  $2 x - 4 \sqrt{2} \log{\left(x + 2 \sqrt{2} \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{2 x}{\left(\sqrt{x}\right)^{2} + \sqrt{8}} = \frac{2 x}{x + 2 \sqrt{2}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 x}{x + 2 \sqrt{2}}\, dx = 2 \int \frac{x}{x + 2 \sqrt{2}}\, dx$
  1. Rewrite the integrand:
    
    $\frac{x}{x + 2 \sqrt{2}} = 1 - \frac{2 \sqrt{2}}{x + 2 \sqrt{2}}$
  2. Integrate term-by-term:
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{2 \sqrt{2}}{x + 2 \sqrt{2}}\right)\, dx = - 2 \sqrt{2} \int \frac{1}{x + 2 \sqrt{2}}\, dx$
      
      Let $u = x + 2 \sqrt{2}$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x + 2 \sqrt{2} \right)}$
      
      So, the result is: $- 2 \sqrt{2} \log{\left(x + 2 \sqrt{2} \right)}$
    The result is: $x - 2 \sqrt{2} \log{\left(x + 2 \sqrt{2} \right)}$
  So, the result is: $2 x - 4 \sqrt{2} \log{\left(x + 2 \sqrt{2} \right)}$
Method #3
1. Rewrite the integrand:
  
  $\frac{2 x}{\left(\sqrt{x}\right)^{2} + \sqrt{8}} = \frac{2 x}{x + 2 \sqrt{2}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 x}{x + 2 \sqrt{2}}\, dx = 2 \int \frac{x}{x + 2 \sqrt{2}}\, dx$
  1. Rewrite the integrand:
    
    $\frac{x}{x + 2 \sqrt{2}} = 1 - \frac{2 \sqrt{2}}{x + 2 \sqrt{2}}$
  2. Integrate term-by-term:
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{2 \sqrt{2}}{x + 2 \sqrt{2}}\right)\, dx = - 2 \sqrt{2} \int \frac{1}{x + 2 \sqrt{2}}\, dx$
      
      Let $u = x + 2 \sqrt{2}$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x + 2 \sqrt{2} \right)}$
      
      So, the result is: $- 2 \sqrt{2} \log{\left(x + 2 \sqrt{2} \right)}$
    The result is: $x - 2 \sqrt{2} \log{\left(x + 2 \sqrt{2} \right)}$
  So, the result is: $2 x - 4 \sqrt{2} \log{\left(x + 2 \sqrt{2} \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                      
 |                                                       
 |      2*x                          ___    /        ___\
 | -------------- dx = C + 2*x - 4*\/ 2 *log\x + 2*\/ 2 /
 |      2                                                
 |   ___      ___                                        
 | \/ x   + \/ 8                                         
 |                                                       
/

$$\int \frac{2 x}{\left(\sqrt{x}\right)^{2} + \sqrt{8}}\, dx = C + 2 x - 4 \sqrt{2} \log{\left(x + 2 \sqrt{2} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

        ___    /        ___\       ___    /        ___\
4 - 4*\/ 2 *log\4 + 2*\/ 2 / + 4*\/ 2 *log\2 + 2*\/ 2 /

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

        ___    /        ___\       ___    /        ___\
4 - 4*\/ 2 *log\4 + 2*\/ 2 / + 4*\/ 2 *log\2 + 2*\/ 2 /

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

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

Numerical answer [src]

2.03948371306291

2.03948371306291

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

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How to use it?

Identical expressions

Similar expressions

Integral of 2x/(✓x²+✓8)dx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3