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Solving integrals/ (x^2-(1/x)^2)п

Integral of (x^2-(1/x)^2)п dx

The solution

You have entered [src]

  2                    
  /                    
 |                     
 |  /          2\      
 |  | 2   /  1\ |      
 |  |x  - |1*-| |*pi dx
 |  \     \  x/ /      
 |                     
/                      
1/2

$$\int\limits_{\frac{1}{2}}^{2} \pi \left(- \left(1 \cdot \frac{1}{x}\right)^{2} + x^{2}\right)\, dx$$

Simplify

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \pi \left(- \left(1 \cdot \frac{1}{x}\right)^{2} + x^{2}\right)\, dx = \pi \int \left(- \left(1 \cdot \frac{1}{x}\right)^{2} + x^{2}\right)\, dx$
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \left(1 \cdot \frac{1}{x}\right)^{2}\right)\, dx = - \int \left(1 \cdot \frac{1}{x}\right)^{2}\, dx$
    1. Let $u = 1 \cdot \frac{1}{x}$ .
      
      Then let $du = - \frac{dx}{x^{2}}$ and substitute $- du$ :
      
      $\int 1\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \left(-1\right)\, du = - \int 1\, du$
        
        The integral of a constant is the constant times the variable of integration:
        
        $\int 1\, du = u$
        
        So, the result is: $- u$
      Now substitute $u$ back in:
      
      $- \frac{1}{x}$
    So, the result is: $\frac{1}{x}$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  The result is: $\frac{x^{3}}{3} + \frac{1}{x}$
So, the result is: $\pi \left(\frac{x^{3}}{3} + \frac{1}{x}\right)$
Now simplify:

$\frac{\pi \left(x^{4} + 3\right)}{3 x}$
Add the constant of integration:

$\frac{\pi \left(x^{4} + 3\right)}{3 x}+ \mathrm{constant}$

The answer is:

$\frac{\pi \left(x^{4} + 3\right)}{3 x}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                     
 |                                      
 | /          2\                /     3\
 | | 2   /  1\ |                |1   x |
 | |x  - |1*-| |*pi dx = C + pi*|- + --|
 | \     \  x/ /                \x   3 /
 |                                      
/

$$\pi\,\left({{x^3}\over{3}}+{{1}\over{x}}\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

9*pi
----
 8

$${{9\,\pi}\over{8}}$$

9*pi
----
 8

$$\frac{9 \pi}{8}$$

Simplify

Numerical answer [src]

3.53429173528852

3.53429173528852

The graph

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

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Integral of (x^2-(1/x)^2)п dx

Limits of integration:

The graph:

Piecewise:

The solution