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Similar expressions
1/(5x-2*sqrt(x))

Integral of 1/(5x+2*sqrt(x)) dx

The solution

You have entered [src]

  9                 
  /                 
 |                  
 |        1         
 |  ------------- dx
 |            ___   
 |  5*x + 2*\/ x    
 |                  
/                   
1

$$\int\limits_{1}^{9} \frac{1}{2 \sqrt{x} + 5 x}\, dx$$

Integral(1/(5*x + 2*sqrt(x)), (x, 1, 9))

Detail solution

Let $u = \sqrt{x}$ .

Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$ :

$\int \frac{2}{5 u + 2}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{5 u + 2}\, du = 2 \int \frac{1}{5 u + 2}\, du$
  1. Let $u = 5 u + 2$ .
    
    Then let $du = 5 du$ and substitute $\frac{du}{5}$ :
    
    $\int \frac{1}{5 u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{5}$
      1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      So, the result is: $\frac{\log{\left(u \right)}}{5}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(5 u + 2 \right)}}{5}$
  So, the result is: $\frac{2 \log{\left(5 u + 2 \right)}}{5}$
Now substitute $u$ back in:

$\frac{2 \log{\left(5 \sqrt{x} + 2 \right)}}{5}$
Add the constant of integration:

$\frac{2 \log{\left(5 \sqrt{x} + 2 \right)}}{5}+ \mathrm{constant}$

The answer is:

$\frac{2 \log{\left(5 \sqrt{x} + 2 \right)}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                         
 |                             /        ___\
 |       1                2*log\2 + 5*\/ x /
 | ------------- dx = C + ------------------
 |           ___                  5         
 | 5*x + 2*\/ x                             
 |                                          
/

$$\int \frac{1}{2 \sqrt{x} + 5 x}\, dx = C + \frac{2 \log{\left(5 \sqrt{x} + 2 \right)}}{5}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  2*log(7/5)   2*log(17/5)
- ---------- + -----------
      5             5

$$- \frac{2 \log{\left(\frac{7}{5} \right)}}{5} + \frac{2 \log{\left(\frac{17}{5} \right)}}{5}$$

  2*log(7/5)   2*log(17/5)
- ---------- + -----------
      5             5

$$- \frac{2 \log{\left(\frac{7}{5} \right)}}{5} + \frac{2 \log{\left(\frac{17}{5} \right)}}{5}$$

-2*log(7/5)/5 + 2*log(17/5)/5

Numerical answer [src]

0.354921278000361

0.354921278000361

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

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Integral of 1/(5x+2*sqrt(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution