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Integral of d{x}:
Integral of x+y
Integral of t
Integral of x/sin(x)
Integral of atan(x)
Identical expressions
(x+ one /2x+ one)-lnx
(x plus 1 divide by 2x plus 1) minus lnx
(x plus one divide by 2x plus one) minus lnx
x+1/2x+1-lnx
(x+1 divide by 2x+1)-lnx
(x+1/2x+1)-lnxdx
Similar expressions
(x-1/2x+1)-lnx
(x+1/2x+1)+lnx
(x+1/2x-1)-lnx

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

The solution

You have entered [src]

  1                        
  /                        
 |                         
 |  /    x             \   
 |  |x + - + 1 - log(x)| dx
 |  \    2             /   
 |                         
/                          
 -1                        
e

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

Integral(x + x/2 + 1 - log(x), (x, exp(-1), 1))

Detail solution

Integrate term-by-term:
1. 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:
      
      $\int \frac{x}{2}\, dx = \frac{\int x\, dx}{2}$
      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: $\frac{x^{2}}{4}$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
    The result is: $\frac{3 x^{2}}{4}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, dx = x$
  The result is: $\frac{3 x^{2}}{4} + x$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \log{\left(x \right)}\right)\, dx = - \int \log{\left(x \right)}\, dx$
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(x \right)} = \log{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .
    
    Then $\operatorname{du}{\left(x \right)} = \frac{1}{x}$ .
    
    To find $v{\left(x \right)}$ :
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
    Now evaluate the sub-integral.
  2. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, dx = x$
  So, the result is: $- x \log{\left(x \right)} + x$
The result is: $\frac{3 x^{2}}{4} - x \log{\left(x \right)} + 2 x$
Now simplify:

$\frac{x \left(3 x - 4 \log{\left(x \right)} + 8\right)}{4}$
Add the constant of integration:

$\frac{x \left(3 x - 4 \log{\left(x \right)} + 8\right)}{4}+ \mathrm{constant}$

The answer is:

$\frac{x \left(3 x - 4 \log{\left(x \right)} + 8\right)}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                   
 |                                        2           
 | /    x             \                3*x            
 | |x + - + 1 - log(x)| dx = C + 2*x + ---- - x*log(x)
 | \    2             /                 4             
 |                                                    
/

$$\int \left(\left(\left(\frac{x}{2} + x\right) + 1\right) - \log{\left(x \right)}\right)\, dx = C + \frac{3 x^{2}}{4} - x \log{\left(x \right)} + 2 x$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                -2
11      -1   3*e  
-- - 3*e   - -----
4              4

$$- \frac{3}{e} - \frac{3}{4 e^{2}} + \frac{11}{4}$$

                -2
11      -1   3*e  
-- - 3*e   - -----
4              4

$$- \frac{3}{e} - \frac{3}{4 e^{2}} + \frac{11}{4}$$

11/4 - 3*exp(-1) - 3*exp(-2)/4

Numerical answer [src]

1.54486021405821

1.54486021405821

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution