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Integral of d{x}:
Integral of x/e^x
Integral of 1/e^x
Integral of e^(-3x)
Integral of x/x^2
Identical expressions
2x/x*sqrt((2x+ one)/2x)
2x divide by x multiply by square root of ((2x plus 1) divide by 2x)
2x divide by x multiply by square root of ((2x plus one) divide by 2x)
2x/x*√((2x+1)/2x)
2x/xsqrt((2x+1)/2x)
2x/xsqrt2x+1/2x
2x divide by x*sqrt((2x+1) divide by 2x)
2x/x*sqrt((2x+1)/2x)dx
Similar expressions
2x/x*sqrt((2x-1)/2x)

Integral of 2x/x*sqrt((2x+1)/2x) dx

The solution

You have entered [src]

  2                       
  /                       
 |                        
 |          ___________   
 |  2*x    / 2*x + 1      
 |  ---*  /  -------*x  dx
 |   x  \/      2         
 |                        
/                         
1

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

Integral(((2*x)/x)*sqrt(((2*x + 1)/2)*x), (x, 1, 2))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{2 x}{x} \sqrt{x \frac{2 x + 1}{2}} = \sqrt{2} \sqrt{2 x^{2} + x}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sqrt{2} \sqrt{2 x^{2} + x}\, dx = \sqrt{2} \int \sqrt{2 x^{2} + x}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\int \sqrt{2 x^{2} + x}\, dx$
  So, the result is: $\sqrt{2} \int \sqrt{2 x^{2} + x}\, dx$
Method #2
1. Rewrite the integrand:
  
  $\frac{2 x}{x} \sqrt{x \frac{2 x + 1}{2}} = 2 \sqrt{x^{2} + \frac{x}{2}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 2 \sqrt{x^{2} + \frac{x}{2}}\, dx = 2 \int \sqrt{x^{2} + \frac{x}{2}}\, dx$
  1. Rewrite the integrand:
    
    $\text{True}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sqrt{2} \sqrt{2 x^{2} + x}}{2}\, dx = \frac{\sqrt{2} \int \sqrt{2 x^{2} + x}\, dx}{2}$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\int \sqrt{2 x^{2} + x}\, dx$
    So, the result is: $\frac{\sqrt{2} \int \sqrt{2 x^{2} + x}\, dx}{2}$
  So, the result is: $\sqrt{2} \int \sqrt{2 x^{2} + x}\, dx$
Now simplify:

$\sqrt{2} \int \sqrt{x \left(2 x + 1\right)}\, dx$
Add the constant of integration:

$\sqrt{2} \int \sqrt{x \left(2 x + 1\right)}\, dx+ \mathrm{constant}$

The answer is:

$\sqrt{2} \int \sqrt{x \left(2 x + 1\right)}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                     /                
 |                                     |                 
 |         ___________                 |    __________   
 | 2*x    / 2*x + 1               ___  |   /        2    
 | ---*  /  -------*x  dx = C + \/ 2 * | \/  x + 2*x   dx
 |  x  \/      2                       |                 
 |                                    /                  
/

$$\int \frac{2 x}{x} \sqrt{x \frac{2 x + 1}{2}}\, dx = C + \sqrt{2} \int \sqrt{2 x^{2} + x}\, dx$$

Simplify

The answer [src]

      ___        /  ___\        /  ___\       ___
  5*\/ 6    acosh\\/ 5 /   acosh\\/ 3 /   9*\/ 5 
- ------- - ------------ + ------------ + -------
     8           8              8            4

$$- \frac{5 \sqrt{6}}{8} - \frac{\operatorname{acosh}{\left(\sqrt{5} \right)}}{8} + \frac{\operatorname{acosh}{\left(\sqrt{3} \right)}}{8} + \frac{9 \sqrt{5}}{4}$$

      ___        /  ___\        /  ___\       ___
  5*\/ 6    acosh\\/ 5 /   acosh\\/ 3 /   9*\/ 5 
- ------- - ------------ + ------------ + -------
     8           8              8            4

$$- \frac{5 \sqrt{6}}{8} - \frac{\operatorname{acosh}{\left(\sqrt{5} \right)}}{8} + \frac{\operatorname{acosh}{\left(\sqrt{3} \right)}}{8} + \frac{9 \sqrt{5}}{4}$$

-5*sqrt(6)/8 - acosh(sqrt(5))/8 + acosh(sqrt(3))/8 + 9*sqrt(5)/4

Numerical answer [src]

3.46304440508526

3.46304440508526

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

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Identical expressions

Similar expressions

Integral of 2x/x*sqrt((2x+1)/2x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2