Mister Exam

Lang:

Solving integrals/ -log(x)

Integral of -log(x) dx

You have entered [src]

  1           
  /           
 |            
 |  -log(x) dx
 |            
/             
0

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

Integral(-log(x), (x, 0, 1))

Detail solution

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: $- x \log{\left(x \right)} + x$
Now simplify:

$x \left(1 - \log{\left(x \right)}\right)$
Add the constant of integration:

$x \left(1 - \log{\left(x \right)}\right)+ \mathrm{constant}$

The answer is:

$x \left(1 - \log{\left(x \right)}\right)+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$1$$

Numerical answer [src]

1.0

1.0

The graph

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