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

Solving integrals/ (x-1)/(5-2x^2)

Integral of (x-1)/(5-2x^2) dx

The solution

You have entered [src]

  1            
  /            
 |             
 |   x - 1     
 |  -------- dx
 |         2   
 |  5 - 2*x    
 |             
/              
0

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

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

Detail solution

Rewrite the integrand:

$\frac{x - 1}{5 - 2 x^{2}} = \frac{x}{5 - 2 x^{2}} - \frac{1}{5 - 2 x^{2}}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x}{5 - 2 x^{2}}\, dx = - \frac{\int \left(- \frac{4 x}{5 - 2 x^{2}}\right)\, dx}{4}$
  1. Let $u = 5 - 2 x^{2}$ .
    
    Then let $du = - 4 x dx$ and substitute $- \frac{du}{4}$ :
    
    $\int \left(- \frac{1}{4 u}\right)\, du$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    Now substitute $u$ back in:
    
    $\log{\left(5 - 2 x^{2} \right)}$
  So, the result is: $- \frac{\log{\left(5 - 2 x^{2} \right)}}{4}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{5 - 2 x^{2}}\right)\, dx = - \int \frac{1}{5 - 2 x^{2}}\, dx$
  So, the result is: $- \begin{cases} \frac{\sqrt{10} \operatorname{acoth}{\left(\frac{\sqrt{10} x}{5} \right)}}{10} & \text{for}\: x^{2} > \frac{5}{2} \\\frac{\sqrt{10} \operatorname{atanh}{\left(\frac{\sqrt{10} x}{5} \right)}}{10} & \text{for}\: x^{2} < \frac{5}{2} \end{cases}$
The result is: $- \begin{cases} \frac{\sqrt{10} \operatorname{acoth}{\left(\frac{\sqrt{10} x}{5} \right)}}{10} & \text{for}\: x^{2} > \frac{5}{2} \\\frac{\sqrt{10} \operatorname{atanh}{\left(\frac{\sqrt{10} x}{5} \right)}}{10} & \text{for}\: x^{2} < \frac{5}{2} \end{cases} - \frac{\log{\left(5 - 2 x^{2} \right)}}{4}$
Now simplify:

$\begin{cases} - \frac{\log{\left(5 - 2 x^{2} \right)}}{4} - \frac{\sqrt{10} \operatorname{acoth}{\left(\frac{\sqrt{10} x}{5} \right)}}{10} & \text{for}\: x^{2} > \frac{5}{2} \\- \frac{\log{\left(5 - 2 x^{2} \right)}}{4} - \frac{\sqrt{10} \operatorname{atanh}{\left(\frac{\sqrt{10} x}{5} \right)}}{10} & \text{for}\: x^{2} < \frac{5}{2} \end{cases}$
Add the constant of integration:

$\begin{cases} - \frac{\log{\left(5 - 2 x^{2} \right)}}{4} - \frac{\sqrt{10} \operatorname{acoth}{\left(\frac{\sqrt{10} x}{5} \right)}}{10} & \text{for}\: x^{2} > \frac{5}{2} \\- \frac{\log{\left(5 - 2 x^{2} \right)}}{4} - \frac{\sqrt{10} \operatorname{atanh}{\left(\frac{\sqrt{10} x}{5} \right)}}{10} & \text{for}\: x^{2} < \frac{5}{2} \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} - \frac{\log{\left(5 - 2 x^{2} \right)}}{4} - \frac{\sqrt{10} \operatorname{acoth}{\left(\frac{\sqrt{10} x}{5} \right)}}{10} & \text{for}\: x^{2} > \frac{5}{2} \\- \frac{\log{\left(5 - 2 x^{2} \right)}}{4} - \frac{\sqrt{10} \operatorname{atanh}{\left(\frac{\sqrt{10} x}{5} \right)}}{10} & \text{for}\: x^{2} < \frac{5}{2} \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

                     //            /    ____\              \                
                     ||  ____      |x*\/ 10 |              |                
                     ||\/ 10 *acoth|--------|              |                
  /                  ||            \   5    /       2      |                
 |                   ||----------------------  for x  > 5/2|      /       2\
 |  x - 1            ||          10                        |   log\5 - 2*x /
 | -------- dx = C - |<                                    | - -------------
 |        2          ||            /    ____\              |         4      
 | 5 - 2*x           ||  ____      |x*\/ 10 |              |                
 |                   ||\/ 10 *atanh|--------|              |                
/                    ||            \   5    /       2      |                
                     ||----------------------  for x  < 5/2|                
                     \\          10                        /

$$\int \frac{x - 1}{5 - 2 x^{2}}\, dx = C - \begin{cases} \frac{\sqrt{10} \operatorname{acoth}{\left(\frac{\sqrt{10} x}{5} \right)}}{10} & \text{for}\: x^{2} > \frac{5}{2} \\\frac{\sqrt{10} \operatorname{atanh}{\left(\frac{\sqrt{10} x}{5} \right)}}{10} & \text{for}\: x^{2} < \frac{5}{2} \end{cases} - \frac{\log{\left(5 - 2 x^{2} \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

/      ____\ /          /  ____\\   /      ____\    /  ____\   /      ____\ /          /       ____\\   /      ____\    /      ____\
|1   \/ 10 | |          |\/ 10 ||   |1   \/ 10 |    |\/ 10 |   |1   \/ 10 | |          |     \/ 10 ||   |1   \/ 10 |    |    \/ 10 |
|- - ------|*|pi*I + log|------|| + |- + ------|*log|------| - |- - ------|*|pi*I + log|-1 + ------|| - |- + ------|*log|1 + ------|
\4     20  / \          \  2   //   \4     20  /    \  2   /   \4     20  / \          \       2   //   \4     20  /    \      2   /

$$- \left(\frac{\sqrt{10}}{20} + \frac{1}{4}\right) \log{\left(1 + \frac{\sqrt{10}}{2} \right)} + \left(\frac{\sqrt{10}}{20} + \frac{1}{4}\right) \log{\left(\frac{\sqrt{10}}{2} \right)} - \left(\frac{1}{4} - \frac{\sqrt{10}}{20}\right) \left(\log{\left(-1 + \frac{\sqrt{10}}{2} \right)} + i \pi\right) + \left(\frac{1}{4} - \frac{\sqrt{10}}{20}\right) \left(\log{\left(\frac{\sqrt{10}}{2} \right)} + i \pi\right)$$

/      ____\ /          /  ____\\   /      ____\    /  ____\   /      ____\ /          /       ____\\   /      ____\    /      ____\
|1   \/ 10 | |          |\/ 10 ||   |1   \/ 10 |    |\/ 10 |   |1   \/ 10 | |          |     \/ 10 ||   |1   \/ 10 |    |    \/ 10 |
|- - ------|*|pi*I + log|------|| + |- + ------|*log|------| - |- - ------|*|pi*I + log|-1 + ------|| - |- + ------|*log|1 + ------|
\4     20  / \          \  2   //   \4     20  /    \  2   /   \4     20  / \          \       2   //   \4     20  /    \      2   /

(1/4 - sqrt(10)/20)*(pi*i + log(sqrt(10)/2)) + (1/4 + sqrt(10)/20)*log(sqrt(10)/2) - (1/4 - sqrt(10)/20)*(pi*i + log(-1 + sqrt(10)/2)) - (1/4 + sqrt(10)/20)*log(1 + sqrt(10)/2)

Numerical answer [src]

-0.108040810024892

-0.108040810024892

The graph

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

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Identical expressions

Similar expressions

Integral of (x-1)/(5-2x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution