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Identical expressions
x*ln(x^ four)dx
x multiply by ln(x to the power of 4)dx
x multiply by ln(x to the power of four)dx
x*ln(x4)dx
x*lnx4dx
x*ln(x⁴)dx
xln(x^4)dx
xln(x4)dx
xlnx4dx
xlnx^4dx

Integral of x*ln(x^4)dx dx

The solution

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0

$$\int\limits_{0}^{1} x \log{\left(x^{4} \right)}\, dx$$

Integral(x*log(x^4), (x, 0, 1))

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = \log{\left(x^{4} \right)}$ and let $\operatorname{dv}{\left(x \right)} = x$ .

Then $\operatorname{du}{\left(x \right)} = \frac{4}{x}$ .

To find $v{\left(x \right)}$ :
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x\, dx = \frac{x^{2}}{2}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int 2 x\, dx = 2 \int x\, dx$
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x\, dx = \frac{x^{2}}{2}$
So, the result is: $x^{2}$
Now simplify:

$\frac{x^{2} \left(\log{\left(x^{4} \right)} - 2\right)}{2}$
Add the constant of integration:

$\frac{x^{2} \left(\log{\left(x^{4} \right)} - 2\right)}{2}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(\log{\left(x^{4} \right)} - 2\right)}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                  
 |                          2    / 4\
 |      / 4\           2   x *log\x /
 | x*log\x / dx = C - x  + ----------
 |                             2     
/

$$\int x \log{\left(x^{4} \right)}\, dx = C + \frac{x^{2} \log{\left(x^{4} \right)}}{2} - x^{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-1

$$-1$$

-1

$$-1$$

-1

Numerical answer [src]

-1.0

-1.0

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

Other calculators

How to use it?

Identical expressions

Integral of x*ln(x^4)dx dx

Limits of integration:

The graph:

Piecewise:

The solution