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Identical expressions
(x^ five)*ln(x^ five)
(x to the power of 5) multiply by ln(x to the power of 5)
(x to the power of five) multiply by ln(x to the power of five)
(x5)*ln(x5)
x5*lnx5
(x⁵)*ln(x⁵)
(x^5)ln(x^5)
(x5)ln(x5)
x5lnx5
x^5lnx^5
(x^5)*ln(x^5)dx

Integral of (x^5)*ln(x^5) dx

The solution

You have entered [src]

  1              
  /              
 |               
 |   5    / 5\   
 |  x *log\x / dx
 |               
/                
0

$$\int\limits_{0}^{1} x^{5} \log{\left(x^{5} \right)}\, dx$$

Integral(x^5*log(x^5), (x, 0, 1))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = \log{\left(x^{5} \right)}$ and let $\operatorname{dv}{\left(x \right)} = x^{5}$ .

Then $\operatorname{du}{\left(x \right)} = \frac{5}{x}$ .

To find $v{\left(x \right)}$ :
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x^{5}\, dx = \frac{x^{6}}{6}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{5 x^{5}}{6}\, dx = \frac{5 \int x^{5}\, dx}{6}$
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x^{5}\, dx = \frac{x^{6}}{6}$
So, the result is: $\frac{5 x^{6}}{36}$
Now simplify:

$\frac{x^{6} \left(6 \log{\left(x^{5} \right)} - 5\right)}{36}$
Add the constant of integration:

$\frac{x^{6} \left(6 \log{\left(x^{5} \right)} - 5\right)}{36}+ \mathrm{constant}$

The answer is:

$\frac{x^{6} \left(6 \log{\left(x^{5} \right)} - 5\right)}{36}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                     
 |                        6    6    / 5\
 |  5    / 5\          5*x    x *log\x /
 | x *log\x / dx = C - ---- + ----------
 |                      36        6     
/

$$\int x^{5} \log{\left(x^{5} \right)}\, dx = C + \frac{x^{6} \log{\left(x^{5} \right)}}{6} - \frac{5 x^{6}}{36}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-5/36

$$- \frac{5}{36}$$

-5/36

$$- \frac{5}{36}$$

-5/36

Numerical answer [src]

-0.138888888888889

-0.138888888888889

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

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Identical expressions

Integral of (x^5)*ln(x^5) dx

Limits of integration:

The graph:

Piecewise:

The solution