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Integral of lnx*x^9 dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1             
  /             
 |              
 |          9   
 |  log(x)*x  dx
 |              
/               
0               
$$\int\limits_{0}^{1} x^{9} \log{\left(x \right)}\, dx$$
Integral(log(x)*x^9, (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    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. 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:

    Method #2

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. The integral of is when :

      Now evaluate the sub-integral.

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

      1. The integral of is when :

      So, the result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                   
 |                     10    10       
 |         9          x     x  *log(x)
 | log(x)*x  dx = C - --- + ----------
 |                    100       10    
/                                     
$${{x^{10}\,\log x}\over{10}}-{{x^{10}}\over{100}}$$
The answer [src]
-1/100
$$-{{1}\over{100}}$$
=
=
-1/100
$$- \frac{1}{100}$$
Numerical answer [src]
-0.01
-0.01

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