Mister Exam

Other calculators

Solving integrals/ 2x/x*sqrt((2x+1)/2x)

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  2                       
  /                       
 |                        
 |          ___________   
 |  2*x    / 2*x + 1      
 |  ---*  /  -------*x  dx
 |   x  \/      2         
 |                        
/                         
1
∫122xxx2x+12 dx\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

  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

      2xxx2x+12=22x2+x\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:

      ∫22x2+x dx=2∫2x2+x dx\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

        ∫2x2+x dx\int \sqrt{2 x^{2} + x}\, dx

      So, the result is: 2∫2x2+x dx\sqrt{2} \int \sqrt{2 x^{2} + x}\, dx

    Method #2

    1. Rewrite the integrand:

      2xxx2x+12=2x2+x2\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:

      ∫2x2+x2 dx=2∫x2+x2 dx\int 2 \sqrt{x^{2} + \frac{x}{2}}\, dx = 2 \int \sqrt{x^{2} + \frac{x}{2}}\, dx

      1. Rewrite the integrand:

        True\text{True}

      2. The integral of a constant times a function is the constant times the integral of the function:

        ∫22x2+x2 dx=2∫2x2+x dx2\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

          ∫2x2+x dx\int \sqrt{2 x^{2} + x}\, dx

        So, the result is: 2∫2x2+x dx2\frac{\sqrt{2} \int \sqrt{2 x^{2} + x}\, dx}{2}

      So, the result is: 2∫2x2+x dx\sqrt{2} \int \sqrt{2 x^{2} + x}\, dx

  2. Now simplify:

    2∫x(2x+1) dx\sqrt{2} \int \sqrt{x \left(2 x + 1\right)}\, dx

  3. Add the constant of integration:

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

The answer is:

2∫x(2x+1) dx+constant\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                       |                 
 |                                    /                  
/
∫2xxx2x+12 dx=C+2∫2x2+x dx\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
−568−acosh⁡(5)8+acosh⁡(3)8+954- \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
−568−acosh⁡(5)8+acosh⁡(3)8+954- \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.

    © Mister Exam – Calculator