Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of e^(2*x+1)
Integral of (1+x)/x
Integral of (2x+3)/(2x+1)
Integral of xsin(x)cos(x)
Identical expressions
(four -x^ two)^(three / two)
(4 minus x squared ) to the power of (3 divide by 2)
(four minus x to the power of two) to the power of (three divide by two)
(4-x2)(3/2)
4-x23/2
(4-x²)^(3/2)
(4-x to the power of 2) to the power of (3/2)
4-x^2^3/2
(4-x^2)^(3 divide by 2)
(4-x^2)^(3/2)dx
Similar expressions
(4+x^2)^(3/2)

Solving integrals/ (4-x^2)^(3/2)

Integral of (4-x^2)^(3/2) dx

The solution

You have entered [src]

  1               
  /               
 |                
 |          3/2   
 |  /     2\      
 |  \4 - x /    dx
 |                
/                 
0

$$\int\limits_{0}^{1} \left(- x^{2} + 4\right)^{\frac{3}{2}}\, dx$$

Simplify

Detail solution

Rewrite the integrand:

$\left(4 - x^{2}\right)^{\frac{3}{2}} = - x^{2} \sqrt{4 - x^{2}} + 4 \sqrt{4 - x^{2}}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- x^{2} \sqrt{4 - x^{2}}\right)\, dx = - \int x^{2} \sqrt{4 - x^{2}}\, dx$
  1. Rewrite the integrand:
    
    $x^{2} \sqrt{4 - x^{2}} = \frac{- x^{4} + 4 x^{2}}{\sqrt{4 - x^{2}}}$
    
    SqrtQuadraticDenomRule(a=4, b=0, c=-1, coeffs=[-1, 0, 4, 0, 0], context=(-x**4 + 4*x**2)/sqrt(4 - x**2), symbol=x)
  So, the result is: $- \sqrt{4 - x^{2}} \left(\frac{x^{3}}{4} - \frac{x}{2}\right) - 2 \operatorname{asin}{\left(\frac{x}{2} \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 4 \sqrt{4 - x^{2}}\, dx = 4 \int \sqrt{4 - x^{2}}\, dx$
  So, the result is: $2 x \sqrt{4 - x^{2}} + 8 \operatorname{asin}{\left(\frac{x}{2} \right)}$
The result is: $2 x \sqrt{4 - x^{2}} - \sqrt{4 - x^{2}} \left(\frac{x^{3}}{4} - \frac{x}{2}\right) + 6 \operatorname{asin}{\left(\frac{x}{2} \right)}$
Now simplify:

$- \frac{x^{3} \sqrt{4 - x^{2}}}{4} + \frac{5 x \sqrt{4 - x^{2}}}{2} + 6 \operatorname{asin}{\left(\frac{x}{2} \right)}$
Add the constant of integration:

$- \frac{x^{3} \sqrt{4 - x^{2}}}{4} + \frac{5 x \sqrt{4 - x^{2}}}{2} + 6 \operatorname{asin}{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer is:

$- \frac{x^{3} \sqrt{4 - x^{2}}}{4} + \frac{5 x \sqrt{4 - x^{2}}}{2} + 6 \operatorname{asin}{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                         
 |                                                                          
 |         3/2                         ________ /       3\          ________
 | /     2\                   /x\     /      2  |  x   x |         /      2 
 | \4 - x /    dx = C + 6*asin|-| - \/  4 - x  *|- - + --| + 2*x*\/  4 - x  
 |                            \2/               \  2   4 /                  
/

$${{x\,\left(4-x^2\right)^{{{3}\over{2}}}}\over{4}}+{{3\,x\,\sqrt{4-x ^2}}\over{2}}+6\,\arcsin \left({{x}\over{2}}\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

         ___
     9*\/ 3 
pi + -------
        4

$${{4\,\pi+3^{{{5}\over{2}}}}\over{4}}$$

         ___
     9*\/ 3 
pi + -------
        4

$$\pi + \frac{9 \sqrt{3}}{4}$$

Simplify

Numerical answer [src]

7.03870697061977

7.03870697061977

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (4-x^2)^(3/2) dx

Limits of integration:

The graph:

Piecewise:

The solution