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Integral of d{x}:
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Integral of lnx^2/x
Integral of log(x+1,7)
Integral of x^2*sin^7(x)
Identical expressions
log(x+ one , seven)
logarithm of (x plus 1,7)
logarithm of (x plus one , seven)
logx+1,7
log(x+1,7)dx
Similar expressions
log(x-1,7)

Solving integrals/ log(x+1,7)

Integral of log(x+1,7) dx

The solution

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = x + \frac{17}{10}$ .
  
  Then let $du = dx$ and substitute $du$ :
  
  $\int \log{\left(u \right)}\, du$
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(u \right)} = \log{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = 1$ .
    
    Then $\operatorname{du}{\left(u \right)} = \frac{1}{u}$ .
    
    To find $v{\left(u \right)}$ :
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
    Now evaluate the sub-integral.
  2. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, du = u$
  Now substitute $u$ back in:
  
  $- x + \left(x + \frac{17}{10}\right) \log{\left(x + \frac{17}{10} \right)} - \frac{17}{10}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = \log{\left(x + \frac{17}{10} \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .
  
  Then $\operatorname{du}{\left(x \right)} = \frac{1}{x + \frac{17}{10}}$ .
  
  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. There are multiple ways to do this integral.
  Method #1
  1. Rewrite the integrand:
    
    $\frac{x}{x + \frac{17}{10}} = 1 - \frac{17}{10 x + 17}$
  2. Integrate term-by-term:
    
    The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, dx = x$
    
    The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{17}{10 x + 17}\right)\, dx = - 17 \int \frac{1}{10 x + 17}\, dx$
    
    Let $u = 10 x + 17$ .
    
    Then let $du = 10 dx$ and substitute $\frac{du}{10}$ :
    
    $\int \frac{1}{10 u}\, du$
    
    The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{10}$
    
    The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    
    So, the result is: $\frac{\log{\left(u \right)}}{10}$
    
    Now substitute $u$ back in:
    
    $\frac{\log{\left(10 x + 17 \right)}}{10}$
    
    So, the result is: $- \frac{17 \log{\left(10 x + 17 \right)}}{10}$
    
    The result is: $x - \frac{17 \log{\left(10 x + 17 \right)}}{10}$
  Method #2
  1. Rewrite the integrand:
    
    $\frac{x}{x + \frac{17}{10}} = \frac{10 x}{10 x + 17}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{10 x}{10 x + 17}\, dx = 10 \int \frac{x}{10 x + 17}\, dx$
    
    Rewrite the integrand:
    
    $\frac{x}{10 x + 17} = \frac{1}{10} - \frac{17}{10 \left(10 x + 17\right)}$
    
    Integrate term-by-term:
    
    The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{1}{10}\, dx = \frac{x}{10}$
    
    The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{17}{10 \left(10 x + 17\right)}\right)\, dx = - \frac{17 \int \frac{1}{10 x + 17}\, dx}{10}$
    
    Let $u = 10 x + 17$ .
    
    Then let $du = 10 dx$ and substitute $\frac{du}{10}$ :
    
    $\int \frac{1}{10 u}\, du$
    
    The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{10}$
    
    The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    
    So, the result is: $\frac{\log{\left(u \right)}}{10}$
    
    Now substitute $u$ back in:
    
    $\frac{\log{\left(10 x + 17 \right)}}{10}$
    
    So, the result is: $- \frac{17 \log{\left(10 x + 17 \right)}}{100}$
    
    The result is: $\frac{x}{10} - \frac{17 \log{\left(10 x + 17 \right)}}{100}$
    
    So, the result is: $x - \frac{17 \log{\left(10 x + 17 \right)}}{10}$
Now simplify:

$- x + \frac{\left(10 x + 17\right) \log{\left(x + \frac{17}{10} \right)}}{10} - \frac{17}{10}$
Add the constant of integration:

$- x + \frac{\left(10 x + 17\right) \log{\left(x + \frac{17}{10} \right)}}{10} - \frac{17}{10}+ \mathrm{constant}$

The answer is:

$- x + \frac{\left(10 x + 17\right) \log{\left(x + \frac{17}{10} \right)}}{10} - \frac{17}{10}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                    
 |                                                     
 |    /    17\        17           /    17\    /    17\
 | log|x + --| dx = - -- + C - x + |x + --|*log|x + --|
 |    \    10/        10           \    10/    \    10/
 |                                                     
/

$$\int \log{\left(x + \frac{17}{10} \right)}\, dx = C - x + \left(x + \frac{17}{10}\right) \log{\left(x + \frac{17}{10} \right)} - \frac{17}{10}$$

Simplify

The answer [src]

     17*log(17)   17*log(27)      /27\
-1 - ---------- + ---------- + log|--|
         10           10          \10/

$$- \frac{17 \log{\left(17 \right)}}{10} - 1 + \log{\left(\frac{27}{10} \right)} + \frac{17 \log{\left(27 \right)}}{10}$$

     17*log(17)   17*log(27)      /27\
-1 - ---------- + ---------- + log|--|
         10           10          \10/

$$- \frac{17 \log{\left(17 \right)}}{10} - 1 + \log{\left(\frac{27}{10} \right)} + \frac{17 \log{\left(27 \right)}}{10}$$

-1 - 17*log(17)/10 + 17*log(27)/10 + log(27/10)

Numerical answer [src]

0.779711760322075

0.779711760322075

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

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Integral of log(x+1,7) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #1

Method #2