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ln(x^(one / two))
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Similar expressions
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Solving integrals/ ln(x^(1/2))

Integral of ln(x^(1/2)) dx

The solution

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0

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

Integral(log(sqrt(x)), (x, 0, 1))

Detail solution

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

$\frac{x \left(\log{\left(x \right)} - 1\right)}{2}$
Add the constant of integration:

$\frac{x \left(\log{\left(x \right)} - 1\right)}{2}+ \mathrm{constant}$

The answer is:

$\frac{x \left(\log{\left(x \right)} - 1\right)}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-1/2

$$- \frac{1}{2}$$

-1/2

$$- \frac{1}{2}$$

-1/2

Numerical answer [src]

-0.5

-0.5

The graph

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

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Identical expressions

Similar expressions

Integral of ln(x^(1/2)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #1

Method #2

Method #2