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Identical expressions
(3y- one)raiz(3y^ two -2y)dy
(3y minus 1)raiz(3y squared minus 2y)dy
(3y minus one)raiz(3y to the power of two minus 2y)dy
(3y-1)raiz(3y2-2y)dy
3y-1raiz3y2-2ydy
(3y-1)raiz(3y²-2y)dy
(3y-1)raiz(3y to the power of 2-2y)dy
3y-1raiz3y^2-2ydy
Similar expressions
(3y-1)raiz(3y^2+2y)dy
(3y+1)raiz(3y^2-2y)dy

Solving integrals/ (3y-1)raiz(3y^2-2y)dy

Integral of (3y-1)raiz(3y^2-2y)dy dx

The solution

You have entered [src]

  1                             
  /                             
 |                              
 |               ____________   
 |              /    2          
 |  (3*y - 1)*\/  3*y  - 2*y  dy
 |                              
/                               
0

$$\int\limits_{0}^{1} \left(3 y - 1\right) \sqrt{3 y^{2} - 2 y}\, dy$$

Integral((3*y - 1)*sqrt(3*y^2 - 2*y), (y, 0, 1))

Detail solution

Let $u = 3 y^{2} - 2 y$ .

Then let $du = \left(6 y - 2\right) dy$ and substitute $\frac{du}{2}$ :

$\int \frac{\sqrt{u}}{2}\, du$
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:

$\frac{\left(3 y^{2} - 2 y\right)^{\frac{3}{2}}}{3}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                  
 |                                                3/2
 |              ____________          /   2      \   
 |             /    2                 \3*y  - 2*y/   
 | (3*y - 1)*\/  3*y  - 2*y  dy = C + ---------------
 |                                           3       
/

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

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]

(0.332995638440809 - 0.000202028623140647j)

(0.332995638440809 - 0.000202028623140647j)

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

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Integral of (3y-1)raiz(3y^2-2y)dy dx

Limits of integration:

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Piecewise:

The solution