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Identical expressions
y=lnx-3x- two
y equally lnx minus 3x minus 2
y equally lnx minus 3x minus two
y=lnx-3x-2dx
Similar expressions
y=lnx-3x+2
y=lnx+3x-2

Integral of y=lnx-3x-2 dx

The solution

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  1                      
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 |  (log(x) - 3*x - 2) dx
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0

$$\int\limits_{0}^{1} \left(\left(- 3 x + \log{\left(x \right)}\right) - 2\right)\, dx$$

Integral(log(x) - 3*x - 2, (x, 0, 1))

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:
    
    $\int \left(- 3 x\right)\, dx = - 3 \int x\, dx$
    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: $- \frac{3 x^{2}}{2}$
  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:
    
    $\int 1\, dx = x$
  The result is: $- \frac{3 x^{2}}{2} + x \log{\left(x \right)} - x$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \left(-2\right)\, dx = - 2 x$
The result is: $- \frac{3 x^{2}}{2} + x \log{\left(x \right)} - 3 x$
Now simplify:

$\frac{x \left(- 3 x + 2 \log{\left(x \right)} - 6\right)}{2}$
Add the constant of integration:

$\frac{x \left(- 3 x + 2 \log{\left(x \right)} - 6\right)}{2}+ \mathrm{constant}$

The answer is:

$\frac{x \left(- 3 x + 2 \log{\left(x \right)} - 6\right)}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-9/2

$$- \frac{9}{2}$$

-9/2

$$- \frac{9}{2}$$

-9/2

Numerical answer [src]

-4.5

-4.5

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

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Identical expressions

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Integral of y=lnx-3x-2 dx

Limits of integration:

The graph:

Piecewise:

The solution