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Integral of xln^2(2x^2+7)
Identical expressions
xln^ two (two x^2+ seven)
xln squared (2x squared plus 7)
xln to the power of two (two x squared plus seven)
xln2(2x2+7)
xln22x2+7
xln²(2x²+7)
xln to the power of 2(2x to the power of 2+7)
xln^22x^2+7
xln^2(2x^2+7)dx
Similar expressions
xln^2(2x^2-7)

Solving integrals/ xln^2(2x^2+7)

Integral of xln^2(2x^2+7) dx

The solution

You have entered [src]

  1                    
  /                    
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 |       2/   2    \   
 |  x*log \2*x  + 7/ dx
 |                     
/                      
0

$$\int\limits_{0}^{1} x \log{\left(2 x^{2} + 7 \right)}^{2}\, dx$$

Integral(x*log(2*x^2 + 7)^2, (x, 0, 1))

Detail solution

Let $u = \log{\left(2 x^{2} + 7 \right)}$ .

Then let $du = \frac{4 x dx}{2 x^{2} + 7}$ and substitute $\frac{du}{4}$ :

$\int \frac{u^{2} e^{u}}{4}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int u^{2} e^{u}\, du = \frac{\int u^{2} e^{u}\, du}{4}$
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(u \right)} = u^{2}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
    
    Then $\operatorname{du}{\left(u \right)} = 2 u$ .
    
    To find $v{\left(u \right)}$ :
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    Now evaluate the sub-integral.
  2. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(u \right)} = 2 u$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
    
    Then $\operatorname{du}{\left(u \right)} = 2$ .
    
    To find $v{\left(u \right)}$ :
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    Now evaluate the sub-integral.
  3. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 e^{u}\, du = 2 \int e^{u}\, du$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $2 e^{u}$
  So, the result is: $\frac{u^{2} e^{u}}{4} - \frac{u e^{u}}{2} + \frac{e^{u}}{2}$
Now substitute $u$ back in:

$x^{2} + \frac{\left(2 x^{2} + 7\right) \log{\left(2 x^{2} + 7 \right)}^{2}}{4} - \frac{\left(2 x^{2} + 7\right) \log{\left(2 x^{2} + 7 \right)}}{2} + \frac{7}{2}$
Now simplify:

$x^{2} + \frac{\left(2 x^{2} + 7\right) \log{\left(2 x^{2} + 7 \right)}^{2}}{4} - \frac{\left(2 x^{2} + 7\right) \log{\left(2 x^{2} + 7 \right)}}{2} + \frac{7}{2}$
Add the constant of integration:

$x^{2} + \frac{\left(2 x^{2} + 7\right) \log{\left(2 x^{2} + 7 \right)}^{2}}{4} - \frac{\left(2 x^{2} + 7\right) \log{\left(2 x^{2} + 7 \right)}}{2} + \frac{7}{2}+ \mathrm{constant}$

The answer is:

$x^{2} + \frac{\left(2 x^{2} + 7\right) \log{\left(2 x^{2} + 7 \right)}^{2}}{4} - \frac{\left(2 x^{2} + 7\right) \log{\left(2 x^{2} + 7 \right)}}{2} + \frac{7}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                       
 |                                    /   2    \    /   2    \      2/   2    \ /   2    \
 |      2/   2    \      7        2   \2*x  + 7/*log\2*x  + 7/   log \2*x  + 7/*\2*x  + 7/
 | x*log \2*x  + 7/ dx = - + C + x  - ------------------------ + -------------------------
 |                       2                       2                           4            
/

$$\int x \log{\left(2 x^{2} + 7 \right)}^{2}\, dx = C + x^{2} + \frac{\left(2 x^{2} + 7\right) \log{\left(2 x^{2} + 7 \right)}^{2}}{4} - \frac{\left(2 x^{2} + 7\right) \log{\left(2 x^{2} + 7 \right)}}{2} + \frac{7}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                    2                      2   
    9*log(9)   7*log (7)   7*log(7)   9*log (9)
1 - -------- - --------- + -------- + ---------
       2           4          2           4

$$- \frac{9 \log{\left(9 \right)}}{2} - \frac{7 \log{\left(7 \right)}^{2}}{4} + 1 + \frac{7 \log{\left(7 \right)}}{2} + \frac{9 \log{\left(9 \right)}^{2}}{4}$$

                    2                      2   
    9*log(9)   7*log (7)   7*log(7)   9*log (9)
1 - -------- - --------- + -------- + ---------
       2           4          2           4

$$- \frac{9 \log{\left(9 \right)}}{2} - \frac{7 \log{\left(7 \right)}^{2}}{4} + 1 + \frac{7 \log{\left(7 \right)}}{2} + \frac{9 \log{\left(9 \right)}^{2}}{4}$$

1 - 9*log(9)/2 - 7*log(7)^2/4 + 7*log(7)/2 + 9*log(9)^2/4

Numerical answer [src]

2.15922453165002

2.15922453165002

The graph

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

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Identical expressions

Similar expressions

Integral of xln^2(2x^2+7) dx

Limits of integration:

The graph:

Piecewise:

The solution