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z^3*(sqrt(x))^2dx

Integral of z^3*(sqrt(x))^2 dx

The solution

You have entered [src]

  1             
  /             
 |              
 |          2   
 |   3   ___    
 |  z *\/ x   dx
 |              
/               
0

$$\int\limits_{0}^{1} z^{3} \left(\sqrt{x}\right)^{2}\, dx$$

Integral(z^3*(sqrt(x))^2, (x, 0, 1))

Detail solution

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

$\int z^{3} \left(\sqrt{x}\right)^{2}\, dx = z^{3} \int \left(\sqrt{x}\right)^{2}\, dx$
1. Let $u = \sqrt{x}$ .
  
  Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$ :
  
  $\int 2 u^{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int u^{3}\, du = 2 \int u^{3}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{3}\, du = \frac{u^{4}}{4}$
    So, the result is: $\frac{u^{4}}{2}$
  Now substitute $u$ back in:
  
  $\frac{x^{2}}{2}$
So, the result is: $\frac{x^{2} z^{3}}{2}$
Add the constant of integration:

$\frac{x^{2} z^{3}}{2}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} z^{3}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                        
 |                         
 |         2           2  3
 |  3   ___           x *z 
 | z *\/ x   dx = C + -----
 |                      2  
/

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

Simplify

The answer [src]

 3
z 
--
2

$$\frac{z^{3}}{2}$$

 3
z 
--
2

$$\frac{z^{3}}{2}$$

z^3/2

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

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Integral of z^3*(sqrt(x))^2 dx

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