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

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

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The solution

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

  1. Rewrite the integrand:

    x2x+1=12−12(2x+1)\frac{x}{2 x + 1} = \frac{1}{2} - \frac{1}{2 \left(2 x + 1\right)}

  2. Integrate term-by-term:

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

      ∫12 dx=x2\int \frac{1}{2}\, dx = \frac{x}{2}

    1. The integral of a constant times a function is the constant times the integral of the function:

      ∫(−12(2x+1)) dx=−∫12x+1 dx2\int \left(- \frac{1}{2 \left(2 x + 1\right)}\right)\, dx = - \frac{\int \frac{1}{2 x + 1}\, dx}{2}

      1. Let u=2x+1u = 2 x + 1.

        Then let du=2dxdu = 2 dx and substitute du2\frac{du}{2}:

        ∫12u du\int \frac{1}{2 u}\, du

        1. The integral of a constant times a function is the constant times the integral of the function:

          ∫1u du=∫1u du2\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{2}

          1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

          So, the result is: log⁡(u)2\frac{\log{\left(u \right)}}{2}

        Now substitute uu back in:

        log⁡(2x+1)2\frac{\log{\left(2 x + 1 \right)}}{2}

      So, the result is: −log⁡(2x+1)4- \frac{\log{\left(2 x + 1 \right)}}{4}

    The result is: x2−log⁡(2x+1)4\frac{x}{2} - \frac{\log{\left(2 x + 1 \right)}}{4}

  3. Add the constant of integration:

    x2−log⁡(2x+1)4+constant\frac{x}{2} - \frac{\log{\left(2 x + 1 \right)}}{4}+ \mathrm{constant}

The answer is:

x2−log⁡(2x+1)4+constant\frac{x}{2} - \frac{\log{\left(2 x + 1 \right)}}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                 
 |                                  
 |    x             x   log(1 + 2*x)
 | ------- dx = C + - - ------------
 | 2*x + 1          2        4      
 |                                  
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∫x2x+1 dx=C+x2−log⁡(2x+1)4\int \frac{x}{2 x + 1}\, dx = C + \frac{x}{2} - \frac{\log{\left(2 x + 1 \right)}}{4}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.00.5
Plot the graph f(x)
The answer [src] 
1   log(3)
- - ------
2     4
12−log⁡(3)4\frac{1}{2} - \frac{\log{\left(3 \right)}}{4} 
=
= 
1   log(3)
- - ------
2     4
12−log⁡(3)4\frac{1}{2} - \frac{\log{\left(3 \right)}}{4} 
1/2 - log(3)/4
Numerical answer [src] 
0.225346927832973
0.225346927832973
The graph
Integral of x/(2*x+1) dx

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

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