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Identical expressions
x/(sqrt(2x- three))
x divide by ( square root of (2x minus 3))
x divide by ( square root of (2x minus three))
x/(√(2x-3))
x/sqrt2x-3
x divide by (sqrt(2x-3))
x/(sqrt(2x-3))dx
Similar expressions
x/(sqrt(2x+3))

Integral of x/(sqrt(2x-3)) dx

The solution

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  3               
  /               
 |                
 |       x        
 |  ----------- dx
 |    _________   
 |  \/ 2*x - 3    
 |                
/                 
-1

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

Integral(x/sqrt(2*x - 3), (x, -1, 3))

Detail solution

Let $u = \sqrt{2 x - 3}$ .

Then let $du = \frac{dx}{\sqrt{2 x - 3}}$ and substitute $du$ :

$\int \left(\frac{u^{2}}{2} + \frac{3}{2}\right)\, du$
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{u^{2}}{2}\, du = \frac{\int u^{2}\, du}{2}$
    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: $\frac{u^{3}}{6}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{3}{2}\, du = \frac{3 u}{2}$
  The result is: $\frac{u^{3}}{6} + \frac{3 u}{2}$
Now substitute $u$ back in:

$\frac{\left(2 x - 3\right)^{\frac{3}{2}}}{6} + \frac{3 \sqrt{2 x - 3}}{2}$
Now simplify:

$\frac{\left(x + 3\right) \sqrt{2 x - 3}}{3}$
Add the constant of integration:

$\frac{\left(x + 3\right) \sqrt{2 x - 3}}{3}+ \mathrm{constant}$

The answer is:

$\frac{\left(x + 3\right) \sqrt{2 x - 3}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                ___
    ___   2*I*\/ 5 
2*\/ 3  - ---------
              3

$$2 \sqrt{3} - \frac{2 \sqrt{5} i}{3}$$

                ___
    ___   2*I*\/ 5 
2*\/ 3  - ---------
              3

$$2 \sqrt{3} - \frac{2 \sqrt{5} i}{3}$$

2*sqrt(3) - 2*i*sqrt(5)/3

Numerical answer [src]

(3.25819615073864 - 1.44092451072132j)

(3.25819615073864 - 1.44092451072132j)

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

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Integral of x/(sqrt(2x-3)) dx

Limits of integration:

The graph:

Piecewise:

The solution