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1/(2x^2+3x+1)
Solving integrals/ 1/(2x^2+3x+1)

Integral of 1/(2x^2+3x+1) dx

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The solution

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∫011⋅12x2+3x+1 dx\int\limits_{0}^{1} 1 \cdot \frac{1}{2 x^{2} + 3 x + 1}\, dx 
Integral(1/(2*x^2 + 3*x + 1), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    1⋅12x2+3x+1=22x+1−1x+11 \cdot \frac{1}{2 x^{2} + 3 x + 1} = \frac{2}{2 x + 1} - \frac{1}{x + 1}

  2. Integrate term-by-term:

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

      ∫22x+1 dx=2∫12x+1 dx\int \frac{2}{2 x + 1}\, dx = 2 \int \frac{1}{2 x + 1}\, dx

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

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

        ∫14u du\int \frac{1}{4 u}\, du

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

          ∫12u du=∫1u du2\int \frac{1}{2 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)\log{\left(2 x + 1 \right)}

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

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

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

        Then let du=dxdu = dx and substitute dudu:

        ∫1u du\int \frac{1}{u}\, du

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

        Now substitute uu back in:

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

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

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
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 | 1*-------------- dx = C - log(1 + x) + log(1 + 2*x)
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 |   2*x  + 3*x + 1                                   
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log⁡(2 x+1)−log⁡(x+1)\log \left(2\,x+1\right)-\log \left(x+1\right)
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
Plot the graph f(x)
The answer [src] 
log(3/2)
log⁡3−log⁡2\log 3-\log 2 
=
= 
log(3/2)
log⁡(32)\log{\left(\frac{3}{2} \right)}
Simplify
Numerical answer [src] 
0.405465108108164
0.405465108108164
The graph
Integral of 1/(2x^2+3x+1) dx

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

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