Integral of sqrt(1-2x) dx

The solution

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 1/2              
  /               
 |                
 |    _________   
 |  \/ 1 - 2*x  dx
 |                
/                 
0

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

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

Detail solution

Let $u = 1 - 2 x$ .

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

$\int \left(- \frac{\sqrt{u}}{2}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sqrt{u}\, du = - \frac{\int \sqrt{u}\, du}{2}$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}$
  So, the result is: $- \frac{u^{\frac{3}{2}}}{3}$
Now substitute $u$ back in:

$- \frac{\left(1 - 2 x\right)^{\frac{3}{2}}}{3}$
Add the constant of integration:

$- \frac{\left(1 - 2 x\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer is:

$- \frac{\left(1 - 2 x\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                               3/2
 |   _________          (1 - 2*x)   
 | \/ 1 - 2*x  dx = C - ------------
 |                           3      
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/3

$$\frac{1}{3}$$

1/3

$$\frac{1}{3}$$

1/3

Numerical answer [src]

0.333333333333333

0.333333333333333

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

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Integral of sqrt(1-2x) dx

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The solution