Integral of x/sqrt(x+2) dx

The solution

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  1             
  /             
 |              
 |      x       
 |  --------- dx
 |    _______   
 |  \/ x + 2    
 |              
/               
0

$$\int\limits_{0}^{1} \frac{x}{\sqrt{x + 2}}\, dx$$

Integral(x/sqrt(x + 2), (x, 0, 1))

Detail solution

Let $u = \sqrt{x + 2}$ .

Then let $du = \frac{dx}{2 \sqrt{x + 2}}$ and substitute $du$ :

$\int \left(2 u^{2} - 4\right)\, du$
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 u^{2}\, du = 2 \int u^{2}\, du$
    1. 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:
    
    $\int \left(-4\right)\, du = - 4 u$
  The result is: $\frac{2 u^{3}}{3} - 4 u$
Now substitute $u$ back in:

$\frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} - 4 \sqrt{x + 2}$
Now simplify:

$\frac{2 \left(x - 4\right) \sqrt{x + 2}}{3}$
Add the constant of integration:

$\frac{2 \left(x - 4\right) \sqrt{x + 2}}{3}+ \mathrm{constant}$

The answer is:

$\frac{2 \left(x - 4\right) \sqrt{x + 2}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                             
 |                                           3/2
 |     x                  _______   2*(x + 2)   
 | --------- dx = C - 4*\/ x + 2  + ------------
 |   _______                             3      
 | \/ x + 2                                     
 |                                              
/

$$\int \frac{x}{\sqrt{x + 2}}\, dx = C + \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} - 4 \sqrt{x + 2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                ___
      ___   8*\/ 2 
- 2*\/ 3  + -------
               3

$$- 2 \sqrt{3} + \frac{8 \sqrt{2}}{3}$$

                ___
      ___   8*\/ 2 
- 2*\/ 3  + -------
               3

$$- 2 \sqrt{3} + \frac{8 \sqrt{2}}{3}$$

-2*sqrt(3) + 8*sqrt(2)/3

Numerical answer [src]

0.307134551190499

0.307134551190499

The graph

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

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Integral of x/sqrt(x+2) dx

Limits of integration:

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The solution