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

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

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  1               
  /               
 |                
 |       x        
 |  ----------- dx
 |    _________   
 |  \/ 2*x + 1    
 |                
/                 
0
∫01x2x+1 dx\int\limits_{0}^{1} \frac{x}{\sqrt{2 x + 1}}\, dx 
Integral(x/sqrt(2*x + 1), (x, 0, 1))
Detail solution

  1. Let u=2x+1u = \sqrt{2 x + 1}.

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

    ∫(u22−12) du\int \left(\frac{u^{2}}{2} - \frac{1}{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:

        ∫u22 du=∫u2 du2\int \frac{u^{2}}{2}\, du = \frac{\int u^{2}\, du}{2}

        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: u36\frac{u^{3}}{6}

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

        ∫(−12) du=−u2\int \left(- \frac{1}{2}\right)\, du = - \frac{u}{2}

      The result is: u36−u2\frac{u^{3}}{6} - \frac{u}{2}

    Now substitute uu back in:

    (2x+1)326−2x+12\frac{\left(2 x + 1\right)^{\frac{3}{2}}}{6} - \frac{\sqrt{2 x + 1}}{2}

  2. Now simplify:

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                               
 |                        _________            3/2
 |      x               \/ 2*x + 1    (2*x + 1)   
 | ----------- dx = C - ----------- + ------------
 |   _________               2             6      
 | \/ 2*x + 1                                     
 |                                                
/
∫x2x+1 dx=C+(2x+1)326−2x+12\int \frac{x}{\sqrt{2 x + 1}}\, dx = C + \frac{\left(2 x + 1\right)^{\frac{3}{2}}}{6} - \frac{\sqrt{2 x + 1}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901.0-1.0
Plot the graph f(x)
The answer [src] 
1/3
13\frac{1}{3} 
=
= 
1/3
13\frac{1}{3} 
1/3
Numerical answer [src] 
0.333333333333333
0.333333333333333

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

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