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x/(sqrt(x)+4)
Solving integrals/ x/(sqrt(x)+4)

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

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  1             
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 |  --------- dx
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 |  \/ x  + 4   
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∫01xx+4 dx\int\limits_{0}^{1} \frac{x}{\sqrt{x} + 4}\, dx 
Integral(x/(sqrt(x) + 4), (x, 0, 1))
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^{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^{3}}{u + 4}\, du = 2 \int \frac{u^{3}}{u + 4}\, du

      1. Rewrite the integrand:

        u3u+4=u2−4u+16−64u+4\frac{u^{3}}{u + 4} = u^{2} - 4 u + 16 - \frac{64}{u + 4}

      2. Integrate term-by-term:

        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}

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

          ∫(−4u) du=−4∫u du\int \left(- 4 u\right)\, du = - 4 \int u\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

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

          So, the result is: −2u2- 2 u^{2}

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

          ∫16 du=16u\int 16\, du = 16 u

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

          ∫(−64u+4) du=−64∫1u+4 du\int \left(- \frac{64}{u + 4}\right)\, du = - 64 \int \frac{1}{u + 4}\, du

          1. Let u=u+4u = u + 4.

            Then let du=dudu = du 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⁡(u+4)\log{\left(u + 4 \right)}

          So, the result is: −64log⁡(u+4)- 64 \log{\left(u + 4 \right)}

        The result is: u33−2u2+16u−64log⁡(u+4)\frac{u^{3}}{3} - 2 u^{2} + 16 u - 64 \log{\left(u + 4 \right)}

      So, the result is: 2u33−4u2+32u−128log⁡(u+4)\frac{2 u^{3}}{3} - 4 u^{2} + 32 u - 128 \log{\left(u + 4 \right)}

    Now substitute uu back in:

    2x323+32x−4x−128log⁡(x+4)\frac{2 x^{\frac{3}{2}}}{3} + 32 \sqrt{x} - 4 x - 128 \log{\left(\sqrt{x} + 4 \right)}

  2. Add the constant of integration:

    2x323+32x−4x−128log⁡(x+4)+constant\frac{2 x^{\frac{3}{2}}}{3} + 32 \sqrt{x} - 4 x - 128 \log{\left(\sqrt{x} + 4 \right)}+ \mathrm{constant}

The answer is:

2x323+32x−4x−128log⁡(x+4)+constant\frac{2 x^{\frac{3}{2}}}{3} + 32 \sqrt{x} - 4 x - 128 \log{\left(\sqrt{x} + 4 \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                               
 |                                                             3/2
 |     x                     /      ___\              ___   2*x   
 | --------- dx = C - 128*log\4 + \/ x / - 4*x + 32*\/ x  + ------
 |   ___                                                      3   
 | \/ x  + 4                                                      
 |                                                                
/
∫xx+4 dx=C+2x323+32x−4x−128log⁡(x+4)\int \frac{x}{\sqrt{x} + 4}\, dx = C + \frac{2 x^{\frac{3}{2}}}{3} + 32 \sqrt{x} - 4 x - 128 \log{\left(\sqrt{x} + 4 \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-200200
Plot the graph f(x)
The answer [src] 
86/3 - 128*log(5) + 128*log(4)
−128log⁡(5)+863+128log⁡(4)- 128 \log{\left(5 \right)} + \frac{86}{3} + 128 \log{\left(4 \right)} 
=
= 
86/3 - 128*log(5) + 128*log(4)
−128log⁡(5)+863+128log⁡(4)- 128 \log{\left(5 \right)} + \frac{86}{3} + 128 \log{\left(4 \right)} 
86/3 - 128*log(5) + 128*log(4)
Numerical answer [src] 
0.104292098447818
0.104292098447818
The graph
Integral of x/(sqrt(x)+4) dx

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

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