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

The solution

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

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

Detail solution

Let $u = \sqrt{x}$ .

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

$\int \frac{2 u^{3}}{u + 1}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{u^{3}}{u + 1}\, du = 2 \int \frac{u^{3}}{u + 1}\, du$
  1. Rewrite the integrand:
    
    $\frac{u^{3}}{u + 1} = u^{2} - u + 1 - \frac{1}{u + 1}$
  2. Integrate term-by-term:
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u\right)\, du = - \int u\, du$
      1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int u\, du = \frac{u^{2}}{2}$
      So, the result is: $- \frac{u^{2}}{2}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{u + 1}\right)\, du = - \int \frac{1}{u + 1}\, du$
      1. Let $u = u + 1$ .
        
        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:
        
        $\log{\left(u + 1 \right)}$
      So, the result is: $- \log{\left(u + 1 \right)}$
    The result is: $\frac{u^{3}}{3} - \frac{u^{2}}{2} + u - \log{\left(u + 1 \right)}$
  So, the result is: $\frac{2 u^{3}}{3} - u^{2} + 2 u - 2 \log{\left(u + 1 \right)}$
Now substitute $u$ back in:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                          
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 |     x                       /      ___\       ___   2*x   
 | --------- dx = C - x - 2*log\1 + \/ x / + 2*\/ x  + ------
 |       ___                                             3   
 | 1 + \/ x                                                  
 |                                                           
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

5/3 - 2*log(2)

$$\frac{5}{3} - 2 \log{\left(2 \right)}$$

5/3 - 2*log(2)

$$\frac{5}{3} - 2 \log{\left(2 \right)}$$

5/3 - 2*log(2)

Numerical answer [src]

0.280372305546776

0.280372305546776

The graph

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

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

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The solution