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Integral of xdx/sqrt(x+1) dx

The solution

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  3             
  /             
 |              
 |      x       
 |  --------- dx
 |    _______   
 |  \/ x + 1    
 |              
/               
0

$$\int\limits_{0}^{3} \frac{x}{\sqrt{x + 1}}\, dx$$

Integral(x/sqrt(x + 1), (x, 0, 3))

Detail solution

Let $u = \sqrt{x + 1}$ .

Then let $du = \frac{dx}{2 \sqrt{x + 1}}$ and substitute $du$ :

$\int \left(2 u^{2} - 2\right)\, du$
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:
    
    $\int \left(-2\right)\, du = - 2 u$
  The result is: $\frac{2 u^{3}}{3} - 2 u$
Now substitute $u$ back in:

$\frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} - 2 \sqrt{x + 1}$
Now simplify:

$\frac{2 \left(x - 2\right) \sqrt{x + 1}}{3}$
Add the constant of integration:

$\frac{2 \left(x - 2\right) \sqrt{x + 1}}{3}+ \mathrm{constant}$

The answer is:

$\frac{2 \left(x - 2\right) \sqrt{x + 1}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

8/3

$$\frac{8}{3}$$

8/3

$$\frac{8}{3}$$

8/3

Numerical answer [src]

2.66666666666667

2.66666666666667

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

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Integral of xdx/sqrt(x+1) dx

Limits of integration:

The graph:

Piecewise:

The solution