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Identical expressions
(9x+ five /sqrt(seven -4x-x²))
(9x plus 5 divide by square root of (7 minus 4x minus x²))
(9x plus five divide by square root of (seven minus 4x minus x²))
(9x+5/√(7-4x-x²))
9x+5/sqrt7-4x-x²
(9x+5 divide by sqrt(7-4x-x²))
(9x+5/sqrt(7-4x-x²))dx
Similar expressions
(9x+5/sqrt(7-4x+x²))
(9x-5/sqrt(7-4x-x²))
(9x+5/sqrt(7+4x-x²))

Integral of (9x+5/sqrt(7-4x-x²)) dx

The solution

You have entered [src]

  1                             
  /                             
 |                              
 |  /              5        \   
 |  |9*x + -----------------| dx
 |  |         ______________|   
 |  |        /            2 |   
 |  \      \/  7 - 4*x - x  /   
 |                              
/                               
0

$$\int\limits_{0}^{1} \left(9 x + \frac{5}{\sqrt{- x^{2} + \left(7 - 4 x\right)}}\right)\, dx$$

Integral(9*x + 5/sqrt(7 - 4*x - x^2), (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 9 x\, dx = 9 \int x\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  So, the result is: $\frac{9 x^{2}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{5}{\sqrt{- x^{2} + \left(7 - 4 x\right)}}\, dx = 5 \int \frac{1}{\sqrt{- x^{2} + \left(7 - 4 x\right)}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\int \frac{1}{\sqrt{- x^{2} + \left(7 - 4 x\right)}}\, dx$
  So, the result is: $5 \int \frac{1}{\sqrt{- x^{2} + \left(7 - 4 x\right)}}\, dx$
The result is: $\frac{9 x^{2}}{2} + 5 \int \frac{1}{\sqrt{- x^{2} + \left(7 - 4 x\right)}}\, dx$
Now simplify:

$\frac{9 x^{2}}{2} + 5 \int \frac{1}{\sqrt{- x^{2} - 4 x + 7}}\, dx$
Add the constant of integration:

$\frac{9 x^{2}}{2} + 5 \int \frac{1}{\sqrt{- x^{2} - 4 x + 7}}\, dx+ \mathrm{constant}$

The answer is:

$\frac{9 x^{2}}{2} + 5 \int \frac{1}{\sqrt{- x^{2} - 4 x + 7}}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                       /                           
 |                                       |                           2
 | /              5        \             |         1              9*x 
 | |9*x + -----------------| dx = C + 5* | ----------------- dx + ----
 | |         ______________|             |    ______________       2  
 | |        /            2 |             |   /            2           
 | \      \/  7 - 4*x - x  /             | \/  7 - 4*x - x            
 |                                       |                            
/                                       /

$$\int \left(9 x + \frac{5}{\sqrt{- x^{2} + \left(7 - 4 x\right)}}\right)\, dx = C + \frac{9 x^{2}}{2} + 5 \int \frac{1}{\sqrt{- x^{2} + \left(7 - 4 x\right)}}\, dx$$

Simplify

The answer [src]

  1                             
  /                             
 |                              
 |             ______________   
 |            /      2          
 |  5 + 9*x*\/  7 - x  - 4*x    
 |  ------------------------- dx
 |         ______________       
 |        /      2              
 |      \/  7 - x  - 4*x        
 |                              
/                               
0

$$\int\limits_{0}^{1} \frac{9 x \sqrt{- x^{2} - 4 x + 7} + 5}{\sqrt{- x^{2} - 4 x + 7}}\, dx$$

  1                             
  /                             
 |                              
 |             ______________   
 |            /      2          
 |  5 + 9*x*\/  7 - x  - 4*x    
 |  ------------------------- dx
 |         ______________       
 |        /      2              
 |      \/  7 - x  - 4*x        
 |                              
/                               
0

$$\int\limits_{0}^{1} \frac{9 x \sqrt{- x^{2} - 4 x + 7} + 5}{\sqrt{- x^{2} - 4 x + 7}}\, dx$$

Integral((5 + 9*x*sqrt(7 - x^2 - 4*x))/sqrt(7 - x^2 - 4*x), (x, 0, 1))

Numerical answer [src]

6.91500407887273

6.91500407887273

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

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Integral of (9x+5/sqrt(7-4x-x²)) dx

Limits of integration:

The graph:

Piecewise:

The solution