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Identical expressions
(4x+ one)/(sqrt(2x- three)+ seven)
(4x plus 1) divide by ( square root of (2x minus 3) plus 7)
(4x plus one) divide by ( square root of (2x minus three) plus seven)
(4x+1)/(√(2x-3)+7)
4x+1/sqrt2x-3+7
(4x+1) divide by (sqrt(2x-3)+7)
(4x+1)/(sqrt(2x-3)+7)dx
Similar expressions
(4x+1)/(sqrt(2x+3)+7)
(4x-1)/(sqrt(2x-3)+7)
(4x+1)/(sqrt(2x-3)-7)

Solving integrals/ (4x+1)/(sqrt(2x-3)+7)

Integral of (4x+1)/(sqrt(2x-3)+7) dx

The solution

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  6                   
  /                   
 |                    
 |      4*x + 1       
 |  --------------- dx
 |    _________       
 |  \/ 2*x - 3  + 7   
 |                    
/                     
2

$$\int\limits_{2}^{6} \frac{4 x + 1}{\sqrt{2 x - 3} + 7}\, dx$$

Integral((4*x + 1)/(sqrt(2*x - 3) + 7), (x, 2, 6))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sqrt{2 x - 3}$ .
  
  Then let $du = \frac{dx}{\sqrt{2 x - 3}}$ and substitute $du$ :
  
  $\int \frac{4 u \left(\frac{u^{2}}{2} + \frac{3}{2}\right) + u}{u + 7}\, du$
  1. Rewrite the integrand:
    
    $\frac{4 u \left(\frac{u^{2}}{2} + \frac{3}{2}\right) + u}{u + 7} = 2 u^{2} - 14 u + 105 - \frac{735}{u + 7}$
  2. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 u^{2}\, du = 2 \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $\frac{2 u^{3}}{3}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 14 u\right)\, du = - 14 \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- 7 u^{2}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 105\, du = 105 u$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{735}{u + 7}\right)\, du = - 735 \int \frac{1}{u + 7}\, du$
      
      Let $u = u + 7$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(u + 7 \right)}$
      
      So, the result is: $- 735 \log{\left(u + 7 \right)}$
    The result is: $\frac{2 u^{3}}{3} - 7 u^{2} + 105 u - 735 \log{\left(u + 7 \right)}$
  Now substitute $u$ back in:
  
  $- 14 x + \frac{2 \left(2 x - 3\right)^{\frac{3}{2}}}{3} + 105 \sqrt{2 x - 3} - 735 \log{\left(\sqrt{2 x - 3} + 7 \right)} + 21$
Method #2
1. Rewrite the integrand:
  
  $\frac{4 x + 1}{\sqrt{2 x - 3} + 7} = \frac{4 x}{\sqrt{2 x - 3} + 7} + \frac{1}{\sqrt{2 x - 3} + 7}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{4 x}{\sqrt{2 x - 3} + 7}\, dx = 4 \int \frac{x}{\sqrt{2 x - 3} + 7}\, dx$
    1. Let $u = \sqrt{2 x - 3}$ .
      
      Then let $du = \frac{dx}{\sqrt{2 x - 3}}$ and substitute $du$ :
      
      $\int \frac{u \left(\frac{u^{2}}{2} + \frac{3}{2}\right)}{u + 7}\, du$
      
      Rewrite the integrand:
      
      $\frac{u \left(\frac{u^{2}}{2} + \frac{3}{2}\right)}{u + 7} = \frac{u^{2}}{2} - \frac{7 u}{2} + 26 - \frac{182}{u + 7}$
      
      Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{u^{2}}{2}\, du = \frac{\int u^{2}\, du}{2}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $\frac{u^{3}}{6}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{7 u}{2}\right)\, du = - \frac{7 \int u\, du}{2}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{7 u^{2}}{4}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 26\, du = 26 u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{182}{u + 7}\right)\, du = - 182 \int \frac{1}{u + 7}\, du$
      
      Let $u = u + 7$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(u + 7 \right)}$
      
      So, the result is: $- 182 \log{\left(u + 7 \right)}$
      
      The result is: $\frac{u^{3}}{6} - \frac{7 u^{2}}{4} + 26 u - 182 \log{\left(u + 7 \right)}$
      
      Now substitute $u$ back in:
      
      $- \frac{7 x}{2} + \frac{\left(2 x - 3\right)^{\frac{3}{2}}}{6} + 26 \sqrt{2 x - 3} - 182 \log{\left(\sqrt{2 x - 3} + 7 \right)} + \frac{21}{4}$
    So, the result is: $- 14 x + \frac{2 \left(2 x - 3\right)^{\frac{3}{2}}}{3} + 104 \sqrt{2 x - 3} - 728 \log{\left(\sqrt{2 x - 3} + 7 \right)} + 21$
  1. Let $u = \sqrt{2 x - 3}$ .
    
    Then let $du = \frac{dx}{\sqrt{2 x - 3}}$ and substitute $du$ :
    
    $\int \frac{u}{u + 7}\, du$
    1. Rewrite the integrand:
      
      $\frac{u}{u + 7} = 1 - \frac{7}{u + 7}$
    2. Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{7}{u + 7}\right)\, du = - 7 \int \frac{1}{u + 7}\, du$
      
      Let $u = u + 7$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(u + 7 \right)}$
      
      So, the result is: $- 7 \log{\left(u + 7 \right)}$
      
      The result is: $u - 7 \log{\left(u + 7 \right)}$
    Now substitute $u$ back in:
    
    $\sqrt{2 x - 3} - 7 \log{\left(\sqrt{2 x - 3} + 7 \right)}$
  The result is: $- 14 x + \frac{2 \left(2 x - 3\right)^{\frac{3}{2}}}{3} + 105 \sqrt{2 x - 3} - 735 \log{\left(\sqrt{2 x - 3} + 7 \right)} + 21$
Now simplify:

$- 14 x + \frac{2 \left(2 x - 3\right)^{\frac{3}{2}}}{3} + 105 \sqrt{2 x - 3} - 735 \log{\left(\sqrt{2 x - 3} + 7 \right)} + 21$
Add the constant of integration:

$- 14 x + \frac{2 \left(2 x - 3\right)^{\frac{3}{2}}}{3} + 105 \sqrt{2 x - 3} - 735 \log{\left(\sqrt{2 x - 3} + 7 \right)} + 21+ \mathrm{constant}$

The answer is:

$- 14 x + \frac{2 \left(2 x - 3\right)^{\frac{3}{2}}}{3} + 105 \sqrt{2 x - 3} - 735 \log{\left(\sqrt{2 x - 3} + 7 \right)} + 21+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                
 |                                                                                              3/2
 |     4*x + 1                          /      _________\                _________   2*(2*x - 3)   
 | --------------- dx = 21 + C - 735*log\7 + \/ 2*x - 3 / - 14*x + 105*\/ 2*x - 3  + --------------
 |   _________                                                                             3       
 | \/ 2*x - 3  + 7                                                                                 
 |                                                                                                 
/

$$\int \frac{4 x + 1}{\sqrt{2 x - 3} + 7}\, dx = C - 14 x + \frac{2 \left(2 x - 3\right)^{\frac{3}{2}}}{3} + 105 \sqrt{2 x - 3} - 735 \log{\left(\sqrt{2 x - 3} + 7 \right)} + 21$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

514/3 - 735*log(10) + 735*log(8)

$$- 735 \log{\left(10 \right)} + \frac{514}{3} + 735 \log{\left(8 \right)}$$

514/3 - 735*log(10) + 735*log(8)

$$- 735 \log{\left(10 \right)} + \frac{514}{3} + 735 \log{\left(8 \right)}$$

514/3 - 735*log(10) + 735*log(8)

Numerical answer [src]

7.32282311738916

7.32282311738916

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

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Identical expressions

Similar expressions

Integral of (4x+1)/(sqrt(2x-3)+7) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2