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Identical expressions
one /x*((sqrt1)+sqrtx)
1 divide by x multiply by (( square root of 1) plus square root of x)
one divide by x multiply by (( square root of 1) plus square root of x)
1/x*((√1)+√x)
1/x((sqrt1)+sqrtx)
1/xsqrt1+sqrtx
1 divide by x*((sqrt1)+sqrtx)
1/x*((sqrt1)+sqrtx)dx
Similar expressions
1/x*((sqrt1)-sqrtx)
1/(x(sqrt(1+(sqrtx))))

Integral of 1/x*((sqrt1)+sqrtx) dx

The solution

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  1                 
  /                 
 |                  
 |    ___     ___   
 |  \/ 1  + \/ x    
 |  ------------- dx
 |        x         
 |                  
/                   
0

$$\int\limits_{0}^{1} \frac{\sqrt{x} + \sqrt{1}}{x}\, dx$$

Integral((sqrt(1) + sqrt(x))/x, (x, 0, 1))

Detail solution

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

$2 \sqrt{x} + \log{\left(x \right)} + 2 \log{\left(2 \right)}$
Add the constant of integration:

$2 \sqrt{x} + \log{\left(x \right)} + 2 \log{\left(2 \right)}+ \mathrm{constant}$

The answer is:

$2 \sqrt{x} + \log{\left(x \right)} + 2 \log{\left(2 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                               
 |                                                
 |   ___     ___                                  
 | \/ 1  + \/ x               ___        /    ___\
 | ------------- dx = C + 2*\/ x  + 2*log\2*\/ x /
 |       x                                        
 |                                                
/

$$\int \frac{\sqrt{x} + \sqrt{1}}{x}\, dx = C + 2 \sqrt{x} + 2 \log{\left(2 \sqrt{x} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

oo

Numerical answer [src]

46.0904461334623

46.0904461334623

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

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Identical expressions

Similar expressions

Integral of 1/x*((sqrt1)+sqrtx) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3