Integral of √x+log2x dx

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  4                      
  /                      
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 |  /  ___           \   
 |  \\/ x  + log(2*x)/ dx
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/                        
1

$$\int\limits_{1}^{4} \left(\sqrt{x} + \log{\left(2 x \right)}\right)\, dx$$

Integral(sqrt(x) + log(2*x), (x, 1, 4))

Detail solution

Integrate term-by-term:
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int \sqrt{x}\, dx = \frac{2 x^{\frac{3}{2}}}{3}$
1. There are multiple ways to do this integral.
  Method #1
  1. Let $u = 2 x$ .
    
    Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{\log{\left(u \right)}}{4}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\log{\left(u \right)}}{2}\, du = \frac{\int \log{\left(u \right)}\, du}{2}$
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(u \right)} = \log{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = 1$ .
      
      Then $\operatorname{du}{\left(u \right)} = \frac{1}{u}$ .
      
      To find $v{\left(u \right)}$ :
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      Now evaluate the sub-integral.
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $\frac{u \log{\left(u \right)}}{2} - \frac{u}{2}$
    Now substitute $u$ back in:
    
    $x \log{\left(2 x \right)} - x$
  Method #2
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(x \right)} = \log{\left(2 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .
    
    Then $\operatorname{du}{\left(x \right)} = \frac{1}{x}$ .
    
    To find $v{\left(x \right)}$ :
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
    Now evaluate the sub-integral.
  2. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, dx = x$
The result is: $\frac{2 x^{\frac{3}{2}}}{3} + x \log{\left(2 x \right)} - x$
Add the constant of integration:

$\frac{2 x^{\frac{3}{2}}}{3} + x \log{\left(2 x \right)} - x+ \mathrm{constant}$

The answer is:

$\frac{2 x^{\frac{3}{2}}}{3} + x \log{\left(2 x \right)} - x+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                   
 |                                    3/2             
 | /  ___           \              2*x                
 | \\/ x  + log(2*x)/ dx = C - x + ------ + x*log(2*x)
 |                                   3                
/

$${{2\,x\,\log \left(2\,x\right)-2\,x}\over{2}}+{{2\,x^{{{3}\over{2}} }}\over{3}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

5/3 - log(2) + 4*log(8)

$${{12\,\log 8-3\,\log 2+5}\over{3}}$$

=

5/3 - log(2) + 4*log(8)

$$- \log{\left(2 \right)} + \frac{5}{3} + 4 \log{\left(8 \right)}$$

Simplify

Numerical answer [src]

9.29128565282607

9.29128565282607

The graph

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Identical expressions

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Integral of √x+log2x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2