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Identical expressions
three /(sqrtx- five)
3 divide by ( square root of x minus 5)
three divide by ( square root of x minus five)
3/(√x-5)
3/sqrtx-5
3 divide by (sqrtx-5)
3/(sqrtx-5)dx
Similar expressions
3/(sqrtx+5)

Integral of 3/(sqrtx-5) dx

The solution

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  1             
  /             
 |              
 |      3       
 |  --------- dx
 |    ___       
 |  \/ x  - 5   
 |              
/               
0

$$\int\limits_{0}^{1} \frac{3}{\sqrt{x} - 5}\, dx$$

Integral(3/(sqrt(x) - 5), (x, 0, 1))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{3}{\sqrt{x} - 5}\, dx = 3 \int \frac{1}{\sqrt{x} - 5}\, dx$
1. Let $u = \sqrt{x}$ .
  
  Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$ :
  
  $\int \frac{2 u}{u - 5}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{u}{u - 5}\, du = 2 \int \frac{u}{u - 5}\, du$
    1. Rewrite the integrand:
      
      $\frac{u}{u - 5} = 1 + \frac{5}{u - 5}$
    2. Integrate term-by-term:
      1. The integral of a constant is the constant times the variable of integration:
        
        $\int 1\, du = u$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \frac{5}{u - 5}\, du = 5 \int \frac{1}{u - 5}\, du$
        
        Let $u = u - 5$ .
        
        Then let $du = du$ and substitute $du$ :
        
        $\int \frac{1}{u}\, du$
        
        The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
        
        Now substitute $u$ back in:
        
        $\log{\left(u - 5 \right)}$
        
        So, the result is: $5 \log{\left(u - 5 \right)}$
      The result is: $u + 5 \log{\left(u - 5 \right)}$
    So, the result is: $2 u + 10 \log{\left(u - 5 \right)}$
  Now substitute $u$ back in:
  
  $2 \sqrt{x} + 10 \log{\left(\sqrt{x} - 5 \right)}$
So, the result is: $6 \sqrt{x} + 30 \log{\left(\sqrt{x} - 5 \right)}$
Add the constant of integration:

$6 \sqrt{x} + 30 \log{\left(\sqrt{x} - 5 \right)}+ \mathrm{constant}$

The answer is:

$6 \sqrt{x} + 30 \log{\left(\sqrt{x} - 5 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                               
 |                                                
 |     3                  ___         /       ___\
 | --------- dx = C + 6*\/ x  + 30*log\-5 + \/ x /
 |   ___                                          
 | \/ x  - 5                                      
 |                                                
/

$$\int \frac{3}{\sqrt{x} - 5}\, dx = C + 6 \sqrt{x} + 30 \log{\left(\sqrt{x} - 5 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

6 - 30*log(5) + 30*log(4)

$$- 30 \log{\left(5 \right)} + 6 + 30 \log{\left(4 \right)}$$

6 - 30*log(5) + 30*log(4)

$$- 30 \log{\left(5 \right)} + 6 + 30 \log{\left(4 \right)}$$

6 - 30*log(5) + 30*log(4)

Numerical answer [src]

-0.694306539426293

-0.694306539426293

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

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Identical expressions

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Integral of 3/(sqrtx-5) dx

Limits of integration:

The graph:

Piecewise:

The solution