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Identical expressions
(6x)/(sqrt(4x+ five))
(6x) divide by ( square root of (4x plus 5))
(6x) divide by ( square root of (4x plus five))
(6x)/(√(4x+5))
6x/sqrt4x+5
(6x) divide by (sqrt(4x+5))
(6x)/(sqrt(4x+5))dx
Similar expressions
(6x)/(sqrt(4x-5))

Integral of (6x)/(sqrt(4x+5)) dx

The solution

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  5               
  /               
 |                
 |      6*x       
 |  ----------- dx
 |    _________   
 |  \/ 4*x + 5    
 |                
/                 
1

$$\int\limits_{1}^{5} \frac{6 x}{\sqrt{4 x + 5}}\, dx$$

Integral((6*x)/sqrt(4*x + 5), (x, 1, 5))

Detail solution

Let $u = \sqrt{4 x + 5}$ .

Then let $du = \frac{2 dx}{\sqrt{4 x + 5}}$ and substitute $du$ :

$\int \left(\frac{3 u^{2}}{4} - \frac{15}{4}\right)\, du$
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{3 u^{2}}{4}\, du = \frac{3 \int u^{2}\, du}{4}$
    1. 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: $\frac{u^{3}}{4}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(- \frac{15}{4}\right)\, du = - \frac{15 u}{4}$
  The result is: $\frac{u^{3}}{4} - \frac{15 u}{4}$
Now substitute $u$ back in:

$\frac{\left(4 x + 5\right)^{\frac{3}{2}}}{4} - \frac{15 \sqrt{4 x + 5}}{4}$
Now simplify:

$\left(x - \frac{5}{2}\right) \sqrt{4 x + 5}$
Add the constant of integration:

$\left(x - \frac{5}{2}\right) \sqrt{4 x + 5}+ \mathrm{constant}$

The answer is:

$\left(x - \frac{5}{2}\right) \sqrt{4 x + 5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                  
 |                           _________            3/2
 |     6*x              15*\/ 4*x + 5    (4*x + 5)   
 | ----------- dx = C - -------------- + ------------
 |   _________                4               4      
 | \/ 4*x + 5                                        
 |                                                   
/

$$\int \frac{6 x}{\sqrt{4 x + 5}}\, dx = C + \frac{\left(4 x + 5\right)^{\frac{3}{2}}}{4} - \frac{15 \sqrt{4 x + 5}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$17$$

Numerical answer [src]

17.0

17.0

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

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Integral of (6x)/(sqrt(4x+5)) dx

Limits of integration:

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Piecewise:

The solution