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Solving integrals/ sqrt^3(x)

Integral of sqrt^3(x) dx

Limits of integration:

from to
v

The graph:

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Piecewise:

The solution

You have entered [src] 
  8          
  /          
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 |       3   
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 |  \/ x   dx
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1
∫18(x)3 dx\int\limits_{1}^{8} \left(\sqrt{x}\right)^{3}\, dx 
Integral((sqrt(x))^3, (x, 1, 8))
Detail solution

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

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

    ∫2u4 du\int 2 u^{4}\, du

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

      ∫u4 du=2∫u4 du\int u^{4}\, du = 2 \int u^{4}\, du

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

        ∫u4 du=u55\int u^{4}\, du = \frac{u^{5}}{5}

      So, the result is: 2u55\frac{2 u^{5}}{5}

    Now substitute uu back in:

    2x525\frac{2 x^{\frac{5}{2}}}{5}

  2. Add the constant of integration:

    2x525+constant\frac{2 x^{\frac{5}{2}}}{5}+ \mathrm{constant}

The answer is:

2x525+constant\frac{2 x^{\frac{5}{2}}}{5}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                      
 |                       
 |      3             5/2
 |   ___           2*x   
 | \/ x   dx = C + ------
 |                   5   
/
∫(x)3 dx=C+2x525\int \left(\sqrt{x}\right)^{3}\, dx = C + \frac{2 x^{\frac{5}{2}}}{5}
Simplify
The graph
1.08.01.52.02.53.03.54.04.55.05.56.06.57.07.50100
Plot the graph f(x)
The answer [src] 
            ___
  2   256*\/ 2 
- - + ---------
  5       5
−25+25625- \frac{2}{5} + \frac{256 \sqrt{2}}{5} 
=
= 
            ___
  2   256*\/ 2 
- - + ---------
  5       5
−25+25625- \frac{2}{5} + \frac{256 \sqrt{2}}{5} 
-2/5 + 256*sqrt(2)/5
Numerical answer [src] 
72.0077343935025
72.0077343935025

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

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