Integral of Xsqrt5+x dx

The solution

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  1                 
  /                 
 |                  
 |  /    ___    \   
 |  \x*\/ 5  + x/ dx
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/                   
0

$$\int\limits_{0}^{1} \left(x + \sqrt{5} x\right)\, dx$$

Integral(x*sqrt(5) + x, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x\, dx = \frac{x^{2}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sqrt{5} x\, dx = \sqrt{5} \int x\, dx$
  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: $\frac{\sqrt{5} x^{2}}{2}$
The result is: $\frac{x^{2}}{2} + \frac{\sqrt{5} x^{2}}{2}$
Now simplify:

$\frac{x^{2} \left(1 + \sqrt{5}\right)}{2}$
Add the constant of integration:

$\frac{x^{2} \left(1 + \sqrt{5}\right)}{2}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(1 + \sqrt{5}\right)}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    
 |                         2     ___  2
 | /    ___    \          x    \/ 5 *x 
 | \x*\/ 5  + x/ dx = C + -- + --------
 |                        2       2    
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

      ___
1   \/ 5 
- + -----
2     2

$$\frac{1}{2} + \frac{\sqrt{5}}{2}$$

      ___
1   \/ 5 
- + -----
2     2

$$\frac{1}{2} + \frac{\sqrt{5}}{2}$$

1/2 + sqrt(5)/2

Numerical answer [src]

1.61803398874989

1.61803398874989

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

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Integral of Xsqrt5+x dx

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The solution