Integral of xln(x+4) dx

The solution

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

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 $\operatorname{du}{\left(x \right)} = \frac{1}{x + 4}$ .

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:

$\int \frac{x^{2}}{2 \left(x + 4\right)}\, dx = \frac{\int \frac{x^{2}}{x + 4}\, dx}{2}$
1. Rewrite the integrand:
  
  $\frac{x^{2}}{x + 4} = x - 4 + \frac{16}{x + 4}$
2. Integrate term-by-term:
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(-4\right)\, dx = - 4 x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{16}{x + 4}\, dx = 16 \int \frac{1}{x + 4}\, dx$
    1. Let $u = x + 4$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      Now substitute $u$ back in:
      
      $\log{\left(x + 4 \right)}$
    So, the result is: $16 \log{\left(x + 4 \right)}$
  The result is: $\frac{x^{2}}{2} - 4 x + 16 \log{\left(x + 4 \right)}$
So, the result is: $\frac{x^{2}}{4} - 2 x + 8 \log{\left(x + 4 \right)}$
Now simplify:

$\frac{x^{2} \log{\left(x + 4 \right)}}{2} - \frac{x^{2}}{4} + 2 x - 8 \log{\left(x + 4 \right)}$
Add the constant of integration:

$\frac{x^{2} \log{\left(x + 4 \right)}}{2} - \frac{x^{2}}{4} + 2 x - 8 \log{\left(x + 4 \right)}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \log{\left(x + 4 \right)}}{2} - \frac{x^{2}}{4} + 2 x - 8 \log{\left(x + 4 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                            2    2           
 |                                            x    x *log(x + 4)
 | x*log(x + 4) dx = C - 8*log(4 + x) + 2*x - -- + -------------
 |                                            4          2      
/

$$\int x \log{\left(x + 4 \right)}\, dx = C + \frac{x^{2} \log{\left(x + 4 \right)}}{2} - \frac{x^{2}}{4} + 2 x - 8 \log{\left(x + 4 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

7              15*log(5)
- + 8*log(4) - ---------
4                  2

$$- \frac{15 \log{\left(5 \right)}}{2} + \frac{7}{4} + 8 \log{\left(4 \right)}$$

7              15*log(5)
- + 8*log(4) - ---------
4                  2

$$- \frac{15 \log{\left(5 \right)}}{2} + \frac{7}{4} + 8 \log{\left(4 \right)}$$

7/4 + 8*log(4) - 15*log(5)/2

Numerical answer [src]

0.769570545703372

0.769570545703372

The graph

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

Other calculators

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Identical expressions

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Integral of xln(x+4) dx

Limits of integration:

The graph:

Piecewise:

The solution