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Identical expressions
two sqrt(3x-2)
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two square root of (3x minus 2)
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2sqrt(3x+2)

Integral of 2sqrt(3x-2) dx

The solution

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  1                 
  /                 
 |                  
 |      _________   
 |  2*\/ 3*x - 2  dx
 |                  
/                   
0

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

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

Detail solution

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

$\int 2 \sqrt{3 x - 2}\, dx = 2 \int \sqrt{3 x - 2}\, dx$
1. Let $u = 3 x - 2$ .
  
  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:
  
  $\frac{2 \left(3 x - 2\right)^{\frac{3}{2}}}{9}$
So, the result is: $\frac{4 \left(3 x - 2\right)^{\frac{3}{2}}}{9}$
Now simplify:

$\frac{4 \left(3 x - 2\right)^{\frac{3}{2}}}{9}$
Add the constant of integration:

$\frac{4 \left(3 x - 2\right)^{\frac{3}{2}}}{9}+ \mathrm{constant}$

The answer is:

$\frac{4 \left(3 x - 2\right)^{\frac{3}{2}}}{9}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                     
 |                                   3/2
 |     _________          4*(3*x - 2)   
 | 2*\/ 3*x - 2  dx = C + --------------
 |                              9       
/

$$\int 2 \sqrt{3 x - 2}\, dx = C + \frac{4 \left(3 x - 2\right)^{\frac{3}{2}}}{9}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

          ___
4   8*I*\/ 2 
- + ---------
9       9

$$\frac{4}{9} + \frac{8 \sqrt{2} i}{9}$$

          ___
4   8*I*\/ 2 
- + ---------
9       9

$$\frac{4}{9} + \frac{8 \sqrt{2} i}{9}$$

4/9 + 8*i*sqrt(2)/9

Numerical answer [src]

(0.443621208120803 + 1.2565791840081j)

(0.443621208120803 + 1.2565791840081j)

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

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Integral of 2sqrt(3x-2) dx

Limits of integration:

The graph:

Piecewise:

The solution