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Integral of (2x-1/x+2/sqrtx) dx

The solution

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  E                     
  /                     
 |                      
 |  /      1     2  \   
 |  |2*x - - + -----| dx
 |  |      x     ___|   
 |  \          \/ x /   
 |                      
/                       
1

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

Integral(2*x - 1/x + 2/sqrt(x), (x, 1, E))

Detail solution

Integrate term-by-term:
1. Integrate term-by-term:
  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: $x^{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{x}\right)\, dx = - \int \frac{1}{x}\, dx$
    1. The integral of $\frac{1}{x}$ is $\log{\left(x \right)}$ .
    So, the result is: $- \log{\left(x \right)}$
  The result is: $x^{2} - \log{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2}{\sqrt{x}}\, dx = 2 \int \frac{1}{\sqrt{x}}\, dx$
  1. Let $u = \sqrt{x}$ .
    
    Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$ :
    
    $\int 2\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      1. The integral of a constant is the constant times the variable of integration:
        
        $\int 1\, du = u$
      So, the result is: $2 u$
    Now substitute $u$ back in:
    
    $2 \sqrt{x}$
  So, the result is: $4 \sqrt{x}$
The result is: $4 \sqrt{x} + x^{2} - \log{\left(x \right)}$
Add the constant of integration:

$4 \sqrt{x} + x^{2} - \log{\left(x \right)}+ \mathrm{constant}$

The answer is:

$4 \sqrt{x} + x^{2} - \log{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                
 |                                                 
 | /      1     2  \           2                ___
 | |2*x - - + -----| dx = C + x  - log(x) + 4*\/ x 
 | |      x     ___|                               
 | \          \/ x /                               
 |                                                 
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

        1/2    2
-6 + 4*e    + e

$$-6 + 4 e^{\frac{1}{2}} + e^{2}$$

        1/2    2
-6 + 4*e    + e

$$-6 + 4 e^{\frac{1}{2}} + e^{2}$$

-6 + 4*exp(1/2) + exp(2)

Numerical answer [src]

7.98394118173116

7.98394118173116

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

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Integral of (2x-1/x+2/sqrtx) dx

Limits of integration:

The graph:

Piecewise:

The solution