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Solving integrals/ dx/(sqrt(3)*sqrt(x)+4)

Integral of dx/(sqrt(3)*sqrt(x)+4) dx

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The solution

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  4                   
  /                   
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 |         1          
 |  --------------- dx
 |    ___   ___       
 |  \/ 3 *\/ x  + 4   
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∫−1413x+4 dx\int\limits_{-1}^{4} \frac{1}{\sqrt{3} \sqrt{x} + 4}\, dx 
Integral(1/(sqrt(3)*sqrt(x) + 4), (x, -1, 4))
Detail solution

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

    Then let du=dx2xdu = \frac{dx}{2 \sqrt{x}} and substitute 2du2 du:

    ∫2u3u+4 du\int \frac{2 u}{\sqrt{3} u + 4}\, du

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

      ∫u3u+4 du=2∫u3u+4 du\int \frac{u}{\sqrt{3} u + 4}\, du = 2 \int \frac{u}{\sqrt{3} u + 4}\, du

      1. Rewrite the integrand:

        u3u+4=33−433(3u+4)\frac{u}{\sqrt{3} u + 4} = \frac{\sqrt{3}}{3} - \frac{4 \sqrt{3}}{3 \left(\sqrt{3} u + 4\right)}

      2. Integrate term-by-term:

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

          ∫33 du=3u3\int \frac{\sqrt{3}}{3}\, du = \frac{\sqrt{3} u}{3}

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

          ∫(−433(3u+4)) du=−43∫13u+4 du3\int \left(- \frac{4 \sqrt{3}}{3 \left(\sqrt{3} u + 4\right)}\right)\, du = - \frac{4 \sqrt{3} \int \frac{1}{\sqrt{3} u + 4}\, du}{3}

          1. Let u=3u+4u = \sqrt{3} u + 4.

            Then let du=3dudu = \sqrt{3} du and substitute 3du3\frac{\sqrt{3} du}{3}:

            ∫33u du\int \frac{\sqrt{3}}{3 u}\, du

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

              ∫1u du=3∫1u du3\int \frac{1}{u}\, du = \frac{\sqrt{3} \int \frac{1}{u}\, du}{3}

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

              So, the result is: 3log⁡(u)3\frac{\sqrt{3} \log{\left(u \right)}}{3}

            Now substitute uu back in:

            3log⁡(3u+4)3\frac{\sqrt{3} \log{\left(\sqrt{3} u + 4 \right)}}{3}

          So, the result is: −4log⁡(3u+4)3- \frac{4 \log{\left(\sqrt{3} u + 4 \right)}}{3}

        The result is: 3u3−4log⁡(3u+4)3\frac{\sqrt{3} u}{3} - \frac{4 \log{\left(\sqrt{3} u + 4 \right)}}{3}

      So, the result is: 23u3−8log⁡(3u+4)3\frac{2 \sqrt{3} u}{3} - \frac{8 \log{\left(\sqrt{3} u + 4 \right)}}{3}

    Now substitute uu back in:

    23x3−8log⁡(3x+4)3\frac{2 \sqrt{3} \sqrt{x}}{3} - \frac{8 \log{\left(\sqrt{3} \sqrt{x} + 4 \right)}}{3}

  2. Add the constant of integration:

    23x3−8log⁡(3x+4)3+constant\frac{2 \sqrt{3} \sqrt{x}}{3} - \frac{8 \log{\left(\sqrt{3} \sqrt{x} + 4 \right)}}{3}+ \mathrm{constant}

The answer is:

23x3−8log⁡(3x+4)3+constant\frac{2 \sqrt{3} \sqrt{x}}{3} - \frac{8 \log{\left(\sqrt{3} \sqrt{x} + 4 \right)}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                               
 |                               /      ___   ___\       ___   ___
 |        1                 8*log\4 + \/ 3 *\/ x /   2*\/ 3 *\/ x 
 | --------------- dx = C - ---------------------- + -------------
 |   ___   ___                        3                    3      
 | \/ 3 *\/ x  + 4                                                
 |                                                                
/
∫13x+4 dx=C+23x3−8log⁡(3x+4)3\int \frac{1}{\sqrt{3} \sqrt{x} + 4}\, dx = C + \frac{2 \sqrt{3} \sqrt{x}}{3} - \frac{8 \log{\left(\sqrt{3} \sqrt{x} + 4 \right)}}{3}
Simplify
The graph
4.00.00.51.01.52.02.53.03.55-5
Plot the graph f(x)
The answer [src] 
       /        ___\       ___        /        ___\         ___
  8*log\4 + 2*\/ 3 /   4*\/ 3    8*log\4 + I*\/ 3 /   2*I*\/ 3 
- ------------------ + ------- + ------------------ - ---------
          3               3              3                3
−8log⁡(23+4)3+433−23i3+8log⁡(4+3i)3- \frac{8 \log{\left(2 \sqrt{3} + 4 \right)}}{3} + \frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3} i}{3} + \frac{8 \log{\left(4 + \sqrt{3} i \right)}}{3} 
=
= 
       /        ___\       ___        /        ___\         ___
  8*log\4 + 2*\/ 3 /   4*\/ 3    8*log\4 + I*\/ 3 /   2*I*\/ 3 
- ------------------ + ------- + ------------------ - ---------
          3               3              3                3
−8log⁡(23+4)3+433−23i3+8log⁡(4+3i)3- \frac{8 \log{\left(2 \sqrt{3} + 4 \right)}}{3} + \frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3} i}{3} + \frac{8 \log{\left(4 + \sqrt{3} i \right)}}{3} 
-8*log(4 + 2*sqrt(3))/3 + 4*sqrt(3)/3 + 8*log(4 + i*sqrt(3))/3 - 2*i*sqrt(3)/3
Numerical answer [src] 
(0.87495726530422 - 0.0650215499529942j)
(0.87495726530422 - 0.0650215499529942j)

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

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