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Identical expressions
(two x- one)/(sqrt5-3x^2)
(2x minus 1) divide by ( square root of 5 minus 3x squared )
(two x minus one) divide by ( square root of 5 minus 3x squared )
(2x-1)/(√5-3x^2)
(2x-1)/(sqrt5-3x2)
2x-1/sqrt5-3x2
(2x-1)/(sqrt5-3x²)
(2x-1)/(sqrt5-3x to the power of 2)
2x-1/sqrt5-3x^2
(2x-1) divide by (sqrt5-3x^2)
(2x-1)/(sqrt5-3x^2)dx
Similar expressions
(2x-1)/(sqrt5+3x^2)
(2x+1)/(sqrt5-3x^2)

Integral of (2x-1)/(sqrt5-3x^2) dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |    2*x - 1      
 |  ------------ dx
 |    ___      2   
 |  \/ 5  - 3*x    
 |                 
/                  
0

$$\int\limits_{0}^{1} \frac{2 x - 1}{- 3 x^{2} + \sqrt{5}}\, dx$$

Integral((2*x - 1)/(sqrt(5) - 3*x^2), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{2 x - 1}{- 3 x^{2} + \sqrt{5}} = \frac{2 x}{- 3 x^{2} + \sqrt{5}} - \frac{1}{- 3 x^{2} + \sqrt{5}}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 x}{- 3 x^{2} + \sqrt{5}}\, dx = - \frac{\int \left(- \frac{6 x}{- 3 x^{2} + \sqrt{5}}\right)\, dx}{3}$
  1. Let $u = - 3 x^{2} + \sqrt{5}$ .
    
    Then let $du = - 6 x dx$ and substitute $- \frac{du}{3}$ :
    
    $\int \left(- \frac{1}{3 u}\right)\, du$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    Now substitute $u$ back in:
    
    $\log{\left(- 3 x^{2} + \sqrt{5} \right)}$
  So, the result is: $- \frac{\log{\left(- 3 x^{2} + \sqrt{5} \right)}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{- 3 x^{2} + \sqrt{5}}\right)\, dx = - \int \frac{1}{- 3 x^{2} + \sqrt{5}}\, dx$
  So, the result is: $- \begin{cases} \frac{\sqrt{3} \cdot 5^{\frac{3}{4}} \operatorname{acoth}{\left(\frac{\sqrt{3} \cdot 5^{\frac{3}{4}} x}{5} \right)}}{15} & \text{for}\: x^{2} > \frac{\sqrt{5}}{3} \\\frac{\sqrt{3} \cdot 5^{\frac{3}{4}} \operatorname{atanh}{\left(\frac{\sqrt{3} \cdot 5^{\frac{3}{4}} x}{5} \right)}}{15} & \text{for}\: x^{2} < \frac{\sqrt{5}}{3} \end{cases}$
The result is: $- \begin{cases} \frac{\sqrt{3} \cdot 5^{\frac{3}{4}} \operatorname{acoth}{\left(\frac{\sqrt{3} \cdot 5^{\frac{3}{4}} x}{5} \right)}}{15} & \text{for}\: x^{2} > \frac{\sqrt{5}}{3} \\\frac{\sqrt{3} \cdot 5^{\frac{3}{4}} \operatorname{atanh}{\left(\frac{\sqrt{3} \cdot 5^{\frac{3}{4}} x}{5} \right)}}{15} & \text{for}\: x^{2} < \frac{\sqrt{5}}{3} \end{cases} - \frac{\log{\left(- 3 x^{2} + \sqrt{5} \right)}}{3}$
Now simplify:

$\begin{cases} - \frac{\log{\left(- 3 x^{2} + \sqrt{5} \right)}}{3} - \frac{\sqrt{3} \cdot 5^{\frac{3}{4}} \operatorname{acoth}{\left(\frac{\sqrt{3} \cdot 5^{\frac{3}{4}} x}{5} \right)}}{15} & \text{for}\: x^{2} > \frac{\sqrt{5}}{3} \\- \frac{\log{\left(- 3 x^{2} + \sqrt{5} \right)}}{3} - \frac{\sqrt{3} \cdot 5^{\frac{3}{4}} \operatorname{atanh}{\left(\frac{\sqrt{3} \cdot 5^{\frac{3}{4}} x}{5} \right)}}{15} & \text{for}\: x^{2} < \frac{\sqrt{5}}{3} \end{cases}$
Add the constant of integration:

$\begin{cases} - \frac{\log{\left(- 3 x^{2} + \sqrt{5} \right)}}{3} - \frac{\sqrt{3} \cdot 5^{\frac{3}{4}} \operatorname{acoth}{\left(\frac{\sqrt{3} \cdot 5^{\frac{3}{4}} x}{5} \right)}}{15} & \text{for}\: x^{2} > \frac{\sqrt{5}}{3} \\- \frac{\log{\left(- 3 x^{2} + \sqrt{5} \right)}}{3} - \frac{\sqrt{3} \cdot 5^{\frac{3}{4}} \operatorname{atanh}{\left(\frac{\sqrt{3} \cdot 5^{\frac{3}{4}} x}{5} \right)}}{15} & \text{for}\: x^{2} < \frac{\sqrt{5}}{3} \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} - \frac{\log{\left(- 3 x^{2} + \sqrt{5} \right)}}{3} - \frac{\sqrt{3} \cdot 5^{\frac{3}{4}} \operatorname{acoth}{\left(\frac{\sqrt{3} \cdot 5^{\frac{3}{4}} x}{5} \right)}}{15} & \text{for}\: x^{2} > \frac{\sqrt{5}}{3} \\- \frac{\log{\left(- 3 x^{2} + \sqrt{5} \right)}}{3} - \frac{\sqrt{3} \cdot 5^{\frac{3}{4}} \operatorname{atanh}{\left(\frac{\sqrt{3} \cdot 5^{\frac{3}{4}} x}{5} \right)}}{15} & \text{for}\: x^{2} < \frac{\sqrt{5}}{3} \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

                         //                /    ___  3/4\                \                    
                         ||  ___  3/4      |x*\/ 3 *5   |                |                    
                         ||\/ 3 *5   *acoth|------------|             ___|                    
  /                      ||                \     5      /       2   \/ 5 |                    
 |                       ||------------------------------  for x  > -----|      /  ___      2\
 |   2*x - 1             ||              15                           3  |   log\\/ 5  - 3*x /
 | ------------ dx = C - |<                                              | - -----------------
 |   ___      2          ||                /    ___  3/4\                |           3        
 | \/ 5  - 3*x           ||  ___  3/4      |x*\/ 3 *5   |                |                    
 |                       ||\/ 3 *5   *atanh|------------|             ___|                    
/                        ||                \     5      /       2   \/ 5 |                    
                         ||------------------------------  for x  < -----|                    
                         \\              15                           3  /

$$\int \frac{2 x - 1}{- 3 x^{2} + \sqrt{5}}\, dx = C - \begin{cases} \frac{\sqrt{3} \cdot 5^{\frac{3}{4}} \operatorname{acoth}{\left(\frac{\sqrt{3} \cdot 5^{\frac{3}{4}} x}{5} \right)}}{15} & \text{for}\: x^{2} > \frac{\sqrt{5}}{3} \\\frac{\sqrt{3} \cdot 5^{\frac{3}{4}} \operatorname{atanh}{\left(\frac{\sqrt{3} \cdot 5^{\frac{3}{4}} x}{5} \right)}}{15} & \text{for}\: x^{2} < \frac{\sqrt{5}}{3} \end{cases} - \frac{\log{\left(- 3 x^{2} + \sqrt{5} \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

nan

$$\text{NaN}$$

nan

$$\text{NaN}$$

nan

Numerical answer [src]

-0.416861480237022

-0.416861480237022

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

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Similar expressions

Integral of (2x-1)/(sqrt5-3x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution