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Identical expressions
sqrt(three *x- one)/ three
square root of (3 multiply by x minus 1) divide by 3
square root of (three multiply by x minus one) divide by three
√(3*x-1)/3
sqrt(3x-1)/3
sqrt3x-1/3
sqrt(3*x-1) divide by 3
sqrt(3*x-1)/3dx
Similar expressions
sqrt(3*x+1)/3

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

The solution

You have entered [src]

  1               
  /               
 |                
 |    _________   
 |  \/ 3*x - 1    
 |  ----------- dx
 |       3        
 |                
/                 
0

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

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

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{\sqrt{3 x - 1}}{3}\, dx = \frac{\int \sqrt{3 x - 1}\, dx}{3}$
1. Let $u = 3 x - 1$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{\sqrt{u}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sqrt{u}\, du = \frac{\int \sqrt{u}\, du}{3}$
    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: $\frac{2 u^{\frac{3}{2}}}{9}$
  Now substitute $u$ back in:
  
  $\frac{2 \left(3 x - 1\right)^{\frac{3}{2}}}{9}$
So, the result is: $\frac{2 \left(3 x - 1\right)^{\frac{3}{2}}}{27}$
Now simplify:

$\frac{2 \left(3 x - 1\right)^{\frac{3}{2}}}{27}$
Add the constant of integration:

$\frac{2 \left(3 x - 1\right)^{\frac{3}{2}}}{27}+ \mathrm{constant}$

The answer is:

$\frac{2 \left(3 x - 1\right)^{\frac{3}{2}}}{27}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                                    
 |   _________                     3/2
 | \/ 3*x - 1           2*(3*x - 1)   
 | ----------- dx = C + --------------
 |      3                     27      
 |                                    
/

$$\int \frac{\sqrt{3 x - 1}}{3}\, dx = C + \frac{2 \left(3 x - 1\right)^{\frac{3}{2}}}{27}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

          ___
2*I   4*\/ 2 
--- + -------
 27      27

$$\frac{4 \sqrt{2}}{27} + \frac{2 i}{27}$$

          ___
2*I   4*\/ 2 
--- + -------
 27      27

$$\frac{4 \sqrt{2}}{27} + \frac{2 i}{27}$$

2*i/27 + 4*sqrt(2)/27

Numerical answer [src]

(0.20942986400135 + 0.0739368680201338j)

(0.20942986400135 + 0.0739368680201338j)

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

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Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution