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x/(sqrt(4*x-5))
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Integral of x/(sqrt(4*x+5)) dx

The solution

You have entered [src]

 pi               
 --               
 2                
  /               
 |                
 |       x        
 |  ----------- dx
 |    _________   
 |  \/ 4*x + 5    
 |                
/                 
0

$$\int\limits_{0}^{\frac{\pi}{2}} \frac{x}{\sqrt{4 x + 5}}\, dx$$

Integral(x/sqrt(4*x + 5), (x, 0, pi/2))

Detail solution

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

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

$\int \left(\frac{u^{2}}{8} - \frac{5}{8}\right)\, du$
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{u^{2}}{8}\, du = \frac{\int u^{2}\, du}{8}$
    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: $\frac{u^{3}}{24}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(- \frac{5}{8}\right)\, du = - \frac{5 u}{8}$
  The result is: $\frac{u^{3}}{24} - \frac{5 u}{8}$
Now substitute $u$ back in:

$\frac{\left(4 x + 5\right)^{\frac{3}{2}}}{24} - \frac{5 \sqrt{4 x + 5}}{8}$
Now simplify:

$\frac{\left(2 x - 5\right) \sqrt{4 x + 5}}{12}$
Add the constant of integration:

$\frac{\left(2 x - 5\right) \sqrt{4 x + 5}}{12}+ \mathrm{constant}$

The answer is:

$\frac{\left(2 x - 5\right) \sqrt{4 x + 5}}{12}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                 
 |                          _________            3/2
 |      x               5*\/ 4*x + 5    (4*x + 5)   
 | ----------- dx = C - ------------- + ------------
 |   _________                8              24     
 | \/ 4*x + 5                                       
 |                                                  
/

$$\int \frac{x}{\sqrt{4 x + 5}}\, dx = C + \frac{\left(4 x + 5\right)^{\frac{3}{2}}}{24} - \frac{5 \sqrt{4 x + 5}}{8}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

      __________       ___        __________
  5*\/ 5 + 2*pi    5*\/ 5    pi*\/ 5 + 2*pi 
- -------------- + ------- + ---------------
        12            12            12

$$- \frac{5 \sqrt{5 + 2 \pi}}{12} + \frac{\pi \sqrt{5 + 2 \pi}}{12} + \frac{5 \sqrt{5}}{12}$$

      __________       ___        __________
  5*\/ 5 + 2*pi    5*\/ 5    pi*\/ 5 + 2*pi 
- -------------- + ------- + ---------------
        12            12            12

$$- \frac{5 \sqrt{5 + 2 \pi}}{12} + \frac{\pi \sqrt{5 + 2 \pi}}{12} + \frac{5 \sqrt{5}}{12}$$

-5*sqrt(5 + 2*pi)/12 + 5*sqrt(5)/12 + pi*sqrt(5 + 2*pi)/12

Numerical answer [src]

0.411488785888517

0.411488785888517

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

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

Limits of integration:

The graph:

Piecewise:

The solution