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Integral of (x+4)/sqrt(9-x^2)
Identical expressions
(x+ four)/sqrt(nine -x^ two)
(x plus 4) divide by square root of (9 minus x squared )
(x plus four) divide by square root of (nine minus x to the power of two)
(x+4)/√(9-x^2)
(x+4)/sqrt(9-x2)
x+4/sqrt9-x2
(x+4)/sqrt(9-x²)
(x+4)/sqrt(9-x to the power of 2)
x+4/sqrt9-x^2
(x+4) divide by sqrt(9-x^2)
(x+4)/sqrt(9-x^2)dx
Similar expressions
(x-4)/sqrt(9-x^2)
(x+4)/sqrt(9+x^2)

Solving integrals/ (x+4)/sqrt(9-x^2)

Integral of (x+4)/sqrt(9-x^2) dx

The solution

You have entered [src]

  3               
  /               
 |                
 |     x + 4      
 |  ----------- dx
 |     ________   
 |    /      2    
 |  \/  9 - x     
 |                
/                 
0

$$\int\limits_{0}^{3} \frac{x + 4}{\sqrt{9 - x^{2}}}\, dx$$

Integral((x + 4)/sqrt(9 - x^2), (x, 0, 3))

Detail solution

Rewrite the integrand:

$\frac{x + 4}{\sqrt{9 - x^{2}}} = \frac{x}{\sqrt{9 - x^{2}}} + \frac{4}{\sqrt{9 - x^{2}}}$
Integrate term-by-term:
1. Let $u = 9 - x^{2}$ .
  
  Then let $du = - 2 x dx$ and substitute $- \frac{du}{2}$ :
  
  $\int \left(- \frac{1}{2 \sqrt{u}}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{\sqrt{u}}\, du = - \frac{\int \frac{1}{\sqrt{u}}\, du}{2}$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{\sqrt{u}}\, du = 2 \sqrt{u}$
    So, the result is: $- \sqrt{u}$
  Now substitute $u$ back in:
  
  $- \sqrt{9 - x^{2}}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{4}{\sqrt{9 - x^{2}}}\, dx = 4 \int \frac{1}{\sqrt{9 - x^{2}}}\, dx$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{\sqrt{9 - x^{2}}}\, dx = \frac{\int \frac{1}{\sqrt{1 - \frac{x^{2}}{9}}}\, dx}{3}$
    1. Let $u = \frac{x}{3}$ .
      
      Then let $du = \frac{dx}{3}$ and substitute $3 du$ :
      
      $\int \frac{9}{\sqrt{1 - u^{2}}}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \frac{3}{\sqrt{1 - u^{2}}}\, du = 3 \int \frac{1}{\sqrt{1 - u^{2}}}\, du$
        
        ArcsinRule(context=1/sqrt(1 - _u**2), symbol=_u)
        
        So, the result is: $3 \operatorname{asin}{\left(u \right)}$
      Now substitute $u$ back in:
      
      $3 \operatorname{asin}{\left(\frac{x}{3} \right)}$
    So, the result is: $\operatorname{asin}{\left(\frac{x}{3} \right)}$
  So, the result is: $4 \operatorname{asin}{\left(\frac{x}{3} \right)}$
The result is: $- \sqrt{9 - x^{2}} + 4 \operatorname{asin}{\left(\frac{x}{3} \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                            
 |                         ________            
 |    x + 4               /      2          /x\
 | ----------- dx = C - \/  9 - x   + 4*asin|-|
 |    ________                              \3/
 |   /      2                                  
 | \/  9 - x                                   
 |                                             
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

3 + 2*pi

$$3 + 2 \pi$$

3 + 2*pi

$$3 + 2 \pi$$

3 + 2*pi

Numerical answer [src]

9.28318530455319

9.28318530455319

The graph

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

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Integral of (x+4)/sqrt(9-x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution