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Integral of d{x}:
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Identical expressions
(x- one)/(x³+ three)dx
(x minus 1) divide by (x³ plus 3)dx
(x minus one) divide by (x³ plus three)dx
x-1/x³+3dx
(x-1) divide by (x³+3)dx
Similar expressions
(x-1)/(x³-3)dx
(x+1)/(x³+3)dx

Integral of (x-1)/(x³+3)dx dx

The solution

You have entered [src]

  1          
  /          
 |           
 |  x - 1    
 |  ------ dx
 |   3       
 |  x  + 3   
 |           
/            
0

$$\int\limits_{0}^{1} \frac{x - 1}{x^{3} + 3}\, dx$$

Integral((x - 1)/(x^3 + 3), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{x - 1}{x^{3} + 3} = \frac{x}{x^{3} + 3} - \frac{1}{x^{3} + 3}$
Integrate term-by-term:
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $- \frac{3^{\frac{2}{3}} \log{\left(x + \sqrt[3]{3} \right)}}{9} + \frac{3^{\frac{2}{3}} \log{\left(x^{2} - \sqrt[3]{3} x + 3^{\frac{2}{3}} \right)}}{18} + \frac{\sqrt[6]{3} \operatorname{atan}{\left(\frac{2 \sqrt[6]{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{x^{3} + 3}\right)\, dx = - \int \frac{1}{x^{3} + 3}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\frac{\sqrt[3]{3} \log{\left(x + \sqrt[3]{3} \right)}}{9} - \frac{\sqrt[3]{3} \log{\left(x^{2} - \sqrt[3]{3} x + 3^{\frac{2}{3}} \right)}}{18} + \frac{3^{\frac{5}{6}} \operatorname{atan}{\left(\frac{2 \sqrt[6]{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{9}$
  So, the result is: $- \frac{\sqrt[3]{3} \log{\left(x + \sqrt[3]{3} \right)}}{9} + \frac{\sqrt[3]{3} \log{\left(x^{2} - \sqrt[3]{3} x + 3^{\frac{2}{3}} \right)}}{18} - \frac{3^{\frac{5}{6}} \operatorname{atan}{\left(\frac{2 \sqrt[6]{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{9}$
The result is: $- \frac{3^{\frac{2}{3}} \log{\left(x + \sqrt[3]{3} \right)}}{9} - \frac{\sqrt[3]{3} \log{\left(x + \sqrt[3]{3} \right)}}{9} + \frac{\sqrt[3]{3} \log{\left(x^{2} - \sqrt[3]{3} x + 3^{\frac{2}{3}} \right)}}{18} + \frac{3^{\frac{2}{3}} \log{\left(x^{2} - \sqrt[3]{3} x + 3^{\frac{2}{3}} \right)}}{18} - \frac{3^{\frac{5}{6}} \operatorname{atan}{\left(\frac{2 \sqrt[6]{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{9} + \frac{\sqrt[6]{3} \operatorname{atan}{\left(\frac{2 \sqrt[6]{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{3}$
Add the constant of integration:

$- \frac{3^{\frac{2}{3}} \log{\left(x + \sqrt[3]{3} \right)}}{9} - \frac{\sqrt[3]{3} \log{\left(x + \sqrt[3]{3} \right)}}{9} + \frac{\sqrt[3]{3} \log{\left(x^{2} - \sqrt[3]{3} x + 3^{\frac{2}{3}} \right)}}{18} + \frac{3^{\frac{2}{3}} \log{\left(x^{2} - \sqrt[3]{3} x + 3^{\frac{2}{3}} \right)}}{18} - \frac{3^{\frac{5}{6}} \operatorname{atan}{\left(\frac{2 \sqrt[6]{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{9} + \frac{\sqrt[6]{3} \operatorname{atan}{\left(\frac{2 \sqrt[6]{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{3}+ \mathrm{constant}$

The answer is:

$- \frac{3^{\frac{2}{3}} \log{\left(x + \sqrt[3]{3} \right)}}{9} - \frac{\sqrt[3]{3} \log{\left(x + \sqrt[3]{3} \right)}}{9} + \frac{\sqrt[3]{3} \log{\left(x^{2} - \sqrt[3]{3} x + 3^{\frac{2}{3}} \right)}}{18} + \frac{3^{\frac{2}{3}} \log{\left(x^{2} - \sqrt[3]{3} x + 3^{\frac{2}{3}} \right)}}{18} - \frac{3^{\frac{5}{6}} \operatorname{atan}{\left(\frac{2 \sqrt[6]{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{9} + \frac{\sqrt[6]{3} \operatorname{atan}{\left(\frac{2 \sqrt[6]{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

                                                                         /    ___       6 ___\             /    ___       6 ___\                                                                 
  /                                                              5/6     |  \/ 3    2*x*\/ 3 |   6 ___     |  \/ 3    2*x*\/ 3 |                                                                 
 |                 3 ___    /    3 ___\    2/3    /    3 ___\   3   *atan|- ----- + ---------|   \/ 3 *atan|- ----- + ---------|   3 ___    / 2/3    2     3 ___\    2/3    / 2/3    2     3 ___\
 | x - 1           \/ 3 *log\x + \/ 3 /   3   *log\x + \/ 3 /            \    3         3    /             \    3         3    /   \/ 3 *log\3    + x  - x*\/ 3 /   3   *log\3    + x  - x*\/ 3 /
 | ------ dx = C - -------------------- - ------------------- - ------------------------------ + ------------------------------- + ------------------------------ + -----------------------------
 |  3                       9                      9                          9                                 3                                18                               18             
 | x  + 3                                                                                                                                                                                        
 |                                                                                                                                                                                               
/

$$\int \frac{x - 1}{x^{3} + 3}\, dx = C - \frac{3^{\frac{2}{3}} \log{\left(x + \sqrt[3]{3} \right)}}{9} - \frac{\sqrt[3]{3} \log{\left(x + \sqrt[3]{3} \right)}}{9} + \frac{\sqrt[3]{3} \log{\left(x^{2} - \sqrt[3]{3} x + 3^{\frac{2}{3}} \right)}}{18} + \frac{3^{\frac{2}{3}} \log{\left(x^{2} - \sqrt[3]{3} x + 3^{\frac{2}{3}} \right)}}{18} - \frac{3^{\frac{5}{6}} \operatorname{atan}{\left(\frac{2 \sqrt[6]{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{9} + \frac{\sqrt[6]{3} \operatorname{atan}{\left(\frac{2 \sqrt[6]{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

         /                             /               2\\          /                             /               2\\
         |     3                       |     9*t   81*t ||          |     3                       |     9*t   81*t ||
- RootSum|243*t  - 27*t + 4, t -> t*log|-3 + --- + -----|| + RootSum|243*t  - 27*t + 4, t -> t*log|-2 + --- + -----||
         \                             \      2      2  //          \                             \      2      2  //

$$- \operatorname{RootSum} {\left(243 t^{3} - 27 t + 4, \left( t \mapsto t \log{\left(\frac{81 t^{2}}{2} + \frac{9 t}{2} - 3 \right)} \right)\right)} + \operatorname{RootSum} {\left(243 t^{3} - 27 t + 4, \left( t \mapsto t \log{\left(\frac{81 t^{2}}{2} + \frac{9 t}{2} - 2 \right)} \right)\right)}$$

         /                             /               2\\          /                             /               2\\
         |     3                       |     9*t   81*t ||          |     3                       |     9*t   81*t ||
- RootSum|243*t  - 27*t + 4, t -> t*log|-3 + --- + -----|| + RootSum|243*t  - 27*t + 4, t -> t*log|-2 + --- + -----||
         \                             \      2      2  //          \                             \      2      2  //

-RootSum(243*_t^3 - 27*_t + 4, Lambda(_t, _t*log(-3 + 9*_t/2 + 81*_t^2/2))) + RootSum(243*_t^3 - 27*_t + 4, Lambda(_t, _t*log(-2 + 9*_t/2 + 81*_t^2/2)))

Numerical answer [src]

-0.161678785119077

-0.161678785119077

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

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Identical expressions

Similar expressions

Integral of (x-1)/(x³+3)dx dx

Limits of integration:

The graph:

Piecewise:

The solution