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Similar expressions
1/(3*x+4*sqrt(x))

Integral of 1/(3x-4sqrt(x)) dx

The solution

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  1                 
  /                 
 |                  
 |        1         
 |  ------------- dx
 |            ___   
 |  3*x - 4*\/ x    
 |                  
/                   
0

$$\int\limits_{0}^{1} \frac{1}{- 4 \sqrt{x} + 3 x}\, dx$$

Integral(1/(3*x - 4*sqrt(x)), (x, 0, 1))

Detail solution

Let $u = \sqrt{x}$ .

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

$\int \frac{2}{3 u - 4}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{3 u - 4}\, du = 2 \int \frac{1}{3 u - 4}\, du$
  1. Let $u = 3 u - 4$ .
    
    Then let $du = 3 du$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{1}{3 u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{3}$
      1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      So, the result is: $\frac{\log{\left(u \right)}}{3}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(3 u - 4 \right)}}{3}$
  So, the result is: $\frac{2 \log{\left(3 u - 4 \right)}}{3}$
Now substitute $u$ back in:

$\frac{2 \log{\left(3 \sqrt{x} - 4 \right)}}{3}$
Add the constant of integration:

$\frac{2 \log{\left(3 \sqrt{x} - 4 \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{2 \log{\left(3 \sqrt{x} - 4 \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                          
 |                             /         ___\
 |       1                2*log\-4 + 3*\/ x /
 | ------------- dx = C + -------------------
 |           ___                   3         
 | 3*x - 4*\/ x                              
 |                                           
/

$$\int \frac{1}{- 4 \sqrt{x} + 3 x}\, dx = C + \frac{2 \log{\left(3 \sqrt{x} - 4 \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  2*log(3)   2*log(4/3)
- -------- - ----------
     3           3

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

  2*log(3)   2*log(4/3)
- -------- - ----------
     3           3

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

-2*log(3)/3 - 2*log(4/3)/3

Numerical answer [src]

-0.924196240579125

-0.924196240579125

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

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Integral of 1/(3x-4sqrt(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution

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Identical expressions

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Integral of 1/(3*x-4*sqrt(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of 1/(3x-4sqrt(x)) dx