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Solving integrals/ xdx/sqrt(x+1)

Integral of xdx/sqrt(x+1) dx

Limits of integration:

from to
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The graph:

from to

Piecewise:

The solution

You have entered [src] 
  3             
  /             
 |              
 |      x       
 |  --------- dx
 |    _______   
 |  \/ x + 1    
 |              
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0
∫03xx+1 dx\int\limits_{0}^{3} \frac{x}{\sqrt{x + 1}}\, dx 
Integral(x/sqrt(x + 1), (x, 0, 3))
Detail solution

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

    Then let du=dx2x+1du = \frac{dx}{2 \sqrt{x + 1}} and substitute dudu:

    ∫(2u2−2) 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:

        ∫2u2 du=2∫u2 du\int 2 u^{2}\, du = 2 \int u^{2}\, du

        1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

          ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

        So, the result is: 2u33\frac{2 u^{3}}{3}

      1. The integral of a constant is the constant times the variable of integration:

        ∫(−2) du=−2u\int \left(-2\right)\, du = - 2 u

      The result is: 2u33−2u\frac{2 u^{3}}{3} - 2 u

    Now substitute uu back in:

    2(x+1)323−2x+1\frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} - 2 \sqrt{x + 1}

  2. Now simplify:

    2(x−2)x+13\frac{2 \left(x - 2\right) \sqrt{x + 1}}{3}

  3. Add the constant of integration:

    2(x−2)x+13+constant\frac{2 \left(x - 2\right) \sqrt{x + 1}}{3}+ \mathrm{constant}

The answer is:

2(x−2)x+13+constant\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                                     
 |                                              
/
∫xx+1 dx=C+2(x+1)323−2x+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
0.003.000.250.500.751.001.251.501.752.002.252.502.755-5
Plot the graph f(x)
The answer [src] 
8/3
83\frac{8}{3} 
=
= 
8/3
83\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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