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Integral of d{x}:
Integral of -e^x
Integral of (x)/(1+x^2)
Integral of dx/(5-3*x)
Integral of dx/(5-2*x)
Identical expressions
dx/(five - three *x)
dx divide by (5 minus 3 multiply by x)
dx divide by (five minus three multiply by x)
dx/(5-3x)
dx/5-3x
dx divide by (5-3*x)
Similar expressions
dx/(5+3*x)

Solving integrals/ dx/(5-3*x)

Integral of dx/(5-3*x) dx

The solution

You have entered [src]

  1           
  /           
 |            
 |     1      
 |  ------- dx
 |  5 - 3*x   
 |            
/             
0

$$\int\limits_{0}^{1} \frac{1}{5 - 3 x}\, dx$$

Integral(1/(5 - 3*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 5 - 3 x$ .
  
  Then let $du = - 3 dx$ and substitute $- \frac{du}{3}$ :
  
  $\int \left(- \frac{1}{3 u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u}\, du = - \frac{\int \frac{1}{u}\, du}{3}$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    So, the result is: $- \frac{\log{\left(u \right)}}{3}$
  Now substitute $u$ back in:
  
  $- \frac{\log{\left(5 - 3 x \right)}}{3}$
Method #2
1. Rewrite the integrand:
  
  $\frac{1}{5 - 3 x} = - \frac{1}{3 x - 5}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{3 x - 5}\right)\, dx = - \int \frac{1}{3 x - 5}\, dx$
  1. Let $u = 3 x - 5$ .
    
    Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{1}{3 u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{3}$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $\frac{\log{\left(u \right)}}{3}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(3 x - 5 \right)}}{3}$
  So, the result is: $- \frac{\log{\left(3 x - 5 \right)}}{3}$
Method #3
1. Rewrite the integrand:
  
  $\frac{1}{5 - 3 x} = - \frac{1}{3 x - 5}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{3 x - 5}\right)\, dx = - \int \frac{1}{3 x - 5}\, dx$
  1. Let $u = 3 x - 5$ .
    
    Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{1}{3 u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{3}$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $\frac{\log{\left(u \right)}}{3}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(3 x - 5 \right)}}{3}$
  So, the result is: $- \frac{\log{\left(3 x - 5 \right)}}{3}$
Add the constant of integration:

$- \frac{\log{\left(5 - 3 x \right)}}{3}+ \mathrm{constant}$

The answer is:

$- \frac{\log{\left(5 - 3 x \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                             
 |                              
 |    1             log(5 - 3*x)
 | ------- dx = C - ------------
 | 5 - 3*x               3      
 |                              
/

$$\int \frac{1}{5 - 3 x}\, dx = C - \frac{\log{\left(5 - 3 x \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  log(2)   log(5)
- ------ + ------
    3        3

$$- \frac{\log{\left(2 \right)}}{3} + \frac{\log{\left(5 \right)}}{3}$$

  log(2)   log(5)
- ------ + ------
    3        3

$$- \frac{\log{\left(2 \right)}}{3} + \frac{\log{\left(5 \right)}}{3}$$

-log(2)/3 + log(5)/3

Numerical answer [src]

0.305430243958052

0.305430243958052

The graph

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

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Identical expressions

Similar expressions

Integral of dx/(5-3*x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3