Mister Exam

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Integral of sqrt(x)*(lnx^2)*x dx

Limits of integration:

from to
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The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                   
  /                   
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 |    ___    2        
 |  \/ x *log (x)*x dx
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$$\int\limits_{0}^{1} \sqrt{x} x \log{\left(x \right)}^{2}\, dx$$
Integral(sqrt(x)*log(x)^2*x, (x, 0, 1))
Detail solution
  1. Let .

    Then let and substitute :

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. There are multiple ways to do this integral.

        Method #1

        1. Let .

          Then let and substitute :

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

            1. The integral of the exponential function is itself.

            So, the result is:

          Now substitute back in:

        Method #2

        1. Let .

          Then let and substitute :

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

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

            So, the result is:

          Now substitute back in:

      Now evaluate the sub-integral.

    2. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. Let .

        Then let and substitute :

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

          1. The integral of the exponential function is itself.

          So, the result is:

        Now substitute back in:

      Now evaluate the sub-integral.

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

      1. Let .

        Then let and substitute :

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

          1. The integral of the exponential function is itself.

          So, the result is:

        Now substitute back in:

      So, the result is:

    Now substitute back in:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                 
 |                              5/2      5/2             5/2    2   
 |   ___    2               16*x      8*x   *log(x)   2*x   *log (x)
 | \/ x *log (x)*x dx = C + ------- - ------------- + --------------
 |                            125           25              5       
/                                                                   
$${{8\,x^{{{5}\over{2}}}\,\left({{25\,\left(\log x\right)^2}\over{4}} -5\,\log x+2\right)}\over{125}}$$
The answer [src]
 16
---
125
$${{16}\over{125}}$$
=
=
 16
---
125
$$\frac{16}{125}$$
Numerical answer [src]
0.128
0.128

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