Integral of ∫2x−−√x−x−−√dx dx

The solution

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 |  /         ___          ___  \   
 |  \2*x - -\/ x  - x - -\/ d *x/ dx
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0

$$\int\limits_{0}^{1} \left(- - \sqrt{d} x - - \sqrt{x} - x + 2 x\right)\, dx$$

Integral(2*x - (-1)*sqrt(x) - x - (-sqrt(d))*x, (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:
  
  $\int \left(- - \sqrt{d} x\right)\, dx = - \int \left(- \sqrt{d} x\right)\, dx$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sqrt{d} x\right)\, dx = - \sqrt{d} \int x\, dx$
    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: $- \frac{\sqrt{d} x^{2}}{2}$
  So, the result is: $\frac{\sqrt{d} x^{2}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \left(-1\right) \sqrt{x}\right)\, dx = - \int \left(- \sqrt{x}\right)\, dx$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sqrt{x}\right)\, dx = - \int \sqrt{x}\, dx$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\frac{2 x^{\frac{3}{2}}}{3}$
    So, the result is: $- \frac{2 x^{\frac{3}{2}}}{3}$
  So, the result is: $\frac{2 x^{\frac{3}{2}}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- x\right)\, dx = - \int x\, dx$
  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: $- \frac{x^{2}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 2 x\, dx = 2 \int x\, dx$
  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: $x^{2}$
The result is: $\frac{\sqrt{d} x^{2}}{2} + \frac{2 x^{\frac{3}{2}}}{3} + \frac{x^{2}}{2}$
Add the constant of integration:

$\frac{\sqrt{d} x^{2}}{2} + \frac{2 x^{\frac{3}{2}}}{3} + \frac{x^{2}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{d} x^{2}}{2} + \frac{2 x^{\frac{3}{2}}}{3} + \frac{x^{2}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                             
 |                                         2      3/2     ___  2
 | /         ___          ___  \          x    2*x      \/ d *x 
 | \2*x - -\/ x  - x - -\/ d *x/ dx = C + -- + ------ + --------
 |                                        2      3         2    
/

$${{\sqrt{d}\,x^2}\over{2}}+{{x^2}\over{2}}+{{2\,x^{{{3}\over{2}}} }\over{3}}$$

Simplify

The answer [src]

      ___
7   \/ d 
- + -----
6     2

$${{3\,\sqrt{d}+7}\over{6}}$$

      ___
7   \/ d 
- + -----
6     2

$$\frac{\sqrt{d}}{2} + \frac{7}{6}$$

Упростить

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

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Integral of ∫2x−−√x−x−−√dx dx

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The solution