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Identical expressions
((two /x)+4x^(- three / two))
((2 divide by x) plus 4x to the power of ( minus 3 divide by 2))
((two divide by x) plus 4x to the power of ( minus three divide by two))
((2/x)+4x(-3/2))
2/x+4x-3/2
2/x+4x^-3/2
((2 divide by x)+4x^(-3 divide by 2))
((2/x)+4x^(-3/2))dx
Similar expressions
((2/x)+4x^(3/2))
((2/x)-4x^(-3/2))

Integral of ((2/x)+4x^(-3/2)) dx

The solution

You have entered [src]

  3              
  /              
 |               
 |  /2    4  \   
 |  |- + ----| dx
 |  |x    3/2|   
 |  \    x   /   
 |               
/                
2

$$\int\limits_{2}^{3} \left(\frac{2}{x} + \frac{4}{x^{\frac{3}{2}}}\right)\, dx$$

Integral(2/x + 4/x^(3/2), (x, 2, 3))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2}{x}\, dx = 2 \int \frac{1}{x}\, dx$
  1. The integral of $\frac{1}{x}$ is $\log{\left(x \right)}$ .
  So, the result is: $2 \log{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{4}{x^{\frac{3}{2}}}\, dx = 4 \int \frac{1}{x^{\frac{3}{2}}}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \frac{1}{x^{\frac{3}{2}}}\, dx = - \frac{2}{\sqrt{x}}$
  So, the result is: $- \frac{8}{\sqrt{x}}$
The result is: $2 \log{\left(x \right)} - \frac{8}{\sqrt{x}}$
Add the constant of integration:

$2 \log{\left(x \right)} - \frac{8}{\sqrt{x}}+ \mathrm{constant}$

The answer is:

$2 \log{\left(x \right)} - \frac{8}{\sqrt{x}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    
 |                                     
 | /2    4  \            8             
 | |- + ----| dx = C - ----- + 2*log(x)
 | |x    3/2|            ___           
 | \    x   /          \/ x            
 |                                     
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                                     ___
                           ___   8*\/ 3 
-2*log(2) + 2*log(3) + 4*\/ 2  - -------
                                    3

$$- \frac{8 \sqrt{3}}{3} - 2 \log{\left(2 \right)} + 2 \log{\left(3 \right)} + 4 \sqrt{2}$$

                                     ___
                           ___   8*\/ 3 
-2*log(2) + 2*log(3) + 4*\/ 2  - -------
                                    3

$$- \frac{8 \sqrt{3}}{3} - 2 \log{\left(2 \right)} + 2 \log{\left(3 \right)} + 4 \sqrt{2}$$

-2*log(2) + 2*log(3) + 4*sqrt(2) - 8*sqrt(3)/3

Numerical answer [src]

1.8489823121917

1.8489823121917

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

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Identical expressions

Similar expressions

Integral of ((2/x)+4x^(-3/2)) dx

Limits of integration:

The graph:

Piecewise:

The solution