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1/(x^2+5x+6)
Solving integrals/ 1/(x^2+5x+6)

Integral of 1/(x^2+5x+6) dx

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The solution

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 |  1*------------ dx
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 |    x  + 5*x + 6   
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∫011⋅1x2+5x+6 dx\int\limits_{0}^{1} 1 \cdot \frac{1}{x^{2} + 5 x + 6}\, dx 
Integral(1/(x^2 + 5*x + 6), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    1⋅1x2+5x+6=−1x+3+1x+21 \cdot \frac{1}{x^{2} + 5 x + 6} = - \frac{1}{x + 3} + \frac{1}{x + 2}

  2. Integrate term-by-term:

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

      ∫(−1x+3) dx=−∫1x+3 dx\int \left(- \frac{1}{x + 3}\right)\, dx = - \int \frac{1}{x + 3}\, dx

      1. Let u=x+3u = x + 3.

        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+3)\log{\left(x + 3 \right)}

      So, the result is: −log⁡(x+3)- \log{\left(x + 3 \right)}

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

      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+2)\log{\left(x + 2 \right)}

    The result is: log⁡(x+2)−log⁡(x+3)\log{\left(x + 2 \right)} - \log{\left(x + 3 \right)}

  3. Add the constant of integration:

    log⁡(x+2)−log⁡(x+3)+constant\log{\left(x + 2 \right)} - \log{\left(x + 3 \right)}+ \mathrm{constant}

The answer is:

log⁡(x+2)−log⁡(x+3)+constant\log{\left(x + 2 \right)} - \log{\left(x + 3 \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                               
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 |        1                                       
 | 1*------------ dx = C - log(3 + x) + log(2 + x)
 |    2                                           
 |   x  + 5*x + 6                                 
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log⁡(x+2)−log⁡(x+3)\log \left(x+2\right)-\log \left(x+3\right)
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.5-0.5
Plot the graph f(x)
The answer [src] 
-log(2) - log(4) + 2*log(3)
−log⁡4+2 log⁡3−log⁡2-\log 4+2\,\log 3-\log 2 
=
= 
-log(2) - log(4) + 2*log(3)
−log⁡(4)−log⁡(2)+2log⁡(3)- \log{\left(4 \right)} - \log{\left(2 \right)} + 2 \log{\left(3 \right)}
Упростить
Numerical answer [src] 
0.117783035656383
0.117783035656383
The graph
Integral of 1/(x^2+5x+6) dx

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

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