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Solving integrals/ x*sqrt(3-x^2)

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

The solution

You have entered [src]

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

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

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

Detail solution

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

Then let $du = - 2 x dx$ 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:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                  
 |                                3/2
 |      ________          /     2\   
 |     /      2           \3 - x /   
 | x*\/  3 - x   dx = C - -----------
 |                             3     
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

            ___
  ___   2*\/ 2 
\/ 3  - -------
           3

$$- \frac{2 \sqrt{2}}{3} + \sqrt{3}$$

            ___
  ___   2*\/ 2 
\/ 3  - -------
           3

$$- \frac{2 \sqrt{2}}{3} + \sqrt{3}$$

sqrt(3) - 2*sqrt(2)/3

Numerical answer [src]

0.789241765986814

0.789241765986814

The graph

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution