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Identical expressions
x/(zero . twenty-five -x^ two)^ zero . five
x divide by (0.25 minus x squared ) to the power of 0.5
x divide by (zero . twenty minus five minus x to the power of two) to the power of zero . five
x/(0.25-x2)0.5
x/0.25-x20.5
x/(0.25-x²)^0.5
x/(0.25-x to the power of 2) to the power of 0.5
x/0.25-x^2^0.5
x divide by (0.25-x^2)^0.5
x/(0.25-x^2)^0.5dx
Similar expressions
x/(0.25+x^2)^0.5

Solving integrals/ x/(0.25-x^2)^0.5

Integral of x/(0.25-x^2)^0.5 dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |       x         
 |  ------------ dx
 |      ________   
 |     / 1    2    
 |    /  - - x     
 |  \/   4         
 |                 
/                  
0

$$\int\limits_{0}^{1} \frac{x}{\sqrt{\frac{1}{4} - x^{2}}}\, dx$$

Integral(x/sqrt(1/4 - x^2), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sqrt{\frac{1}{4} - x^{2}}$ .
  
  Then let $du = - \frac{x dx}{\sqrt{\frac{1}{4} - x^{2}}}$ and substitute $- du$ :
  
  $\int \left(-1\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
    So, the result is: $- u$
  Now substitute $u$ back in:
  
  $- \sqrt{\frac{1}{4} - x^{2}}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x}{\sqrt{\frac{1}{4} - x^{2}}} = \frac{2 x}{\sqrt{1 - 4 x^{2}}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 x}{\sqrt{1 - 4 x^{2}}}\, dx = 2 \int \frac{x}{\sqrt{1 - 4 x^{2}}}\, dx$
  1. Let $u = 1 - 4 x^{2}$ .
    
    Then let $du = - 8 x dx$ and substitute $- \frac{du}{8}$ :
    
    $\int \left(- \frac{1}{8 \sqrt{u}}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{\sqrt{u}}\, du = - \frac{\int \frac{1}{\sqrt{u}}\, du}{8}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{\sqrt{u}}\, du = 2 \sqrt{u}$
      
      So, the result is: $- \frac{\sqrt{u}}{4}$
    Now substitute $u$ back in:
    
    $- \frac{\sqrt{1 - 4 x^{2}}}{4}$
  So, the result is: $- \frac{\sqrt{1 - 4 x^{2}}}{2}$
Now simplify:

$- \frac{\sqrt{1 - 4 x^{2}}}{2}$
Add the constant of integration:

$- \frac{\sqrt{1 - 4 x^{2}}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{\sqrt{1 - 4 x^{2}}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                  
 |                           ________
 |      x                   / 1    2 
 | ------------ dx = C -   /  - - x  
 |     ________          \/   4      
 |    / 1    2                       
 |   /  - - x                        
 | \/   4                            
 |                                   
/

$$\int \frac{x}{\sqrt{\frac{1}{4} - x^{2}}}\, dx = C - \sqrt{\frac{1}{4} - x^{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1                                 
  /                                 
 |                                  
 |  /    -2*I*x             2       
 |  |--------------  for 4*x  > 1   
 |  |   ___________                 
 |  |  /         2                  
 |  |\/  -1 + 4*x                   
 |  <                             dx
 |  |     2*x                       
 |  |-------------    otherwise     
 |  |   __________                  
 |  |  /        2                   
 |  \\/  1 - 4*x                    
 |                                  
/                                   
0

$$\int\limits_{0}^{1} \begin{cases} - \frac{2 i x}{\sqrt{4 x^{2} - 1}} & \text{for}\: 4 x^{2} > 1 \\\frac{2 x}{\sqrt{1 - 4 x^{2}}} & \text{otherwise} \end{cases}\, dx$$

  1                                 
  /                                 
 |                                  
 |  /    -2*I*x             2       
 |  |--------------  for 4*x  > 1   
 |  |   ___________                 
 |  |  /         2                  
 |  |\/  -1 + 4*x                   
 |  <                             dx
 |  |     2*x                       
 |  |-------------    otherwise     
 |  |   __________                  
 |  |  /        2                   
 |  \\/  1 - 4*x                    
 |                                  
/                                   
0

$$\int\limits_{0}^{1} \begin{cases} - \frac{2 i x}{\sqrt{4 x^{2} - 1}} & \text{for}\: 4 x^{2} > 1 \\\frac{2 x}{\sqrt{1 - 4 x^{2}}} & \text{otherwise} \end{cases}\, dx$$

Integral(Piecewise((-2*i*x/sqrt(-1 + 4*x^2), 4*x^2 > 1), (2*x/sqrt(1 - 4*x^2), True)), (x, 0, 1))

The graph

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

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How to use it?

Identical expressions

Similar expressions

Integral of x/(0.25-x^2)^0.5 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2