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Identical expressions
arctan(sqrt(x^ three - one))
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Similar expressions
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Integral of arctan(sqrt(x^3-1)) dx

The solution

You have entered [src]

 oo                     
  /                     
 |                      
 |      /   ________\   
 |      |  /  3     |   
 |  atan\\/  x  - 1 / dx
 |                      
/                       
1

$$\int\limits_{1}^{\infty} \operatorname{atan}{\left(\sqrt{x^{3} - 1} \right)}\, dx$$

Integral(atan(sqrt(x^3 - 1)), (x, 1, oo))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = \operatorname{atan}{\left(\sqrt{x^{3} - 1} \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .

Then $\operatorname{du}{\left(x \right)} = \frac{3}{2 x \sqrt{x^{3} - 1}}$ .

To find $v{\left(x \right)}$ :
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{3}{2 \sqrt{x^{3} - 1}}\, dx = \frac{3 \int \frac{1}{\sqrt{x^{3} - 1}}\, dx}{2}$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $- \frac{i x \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)}}{3 \Gamma\left(\frac{4}{3}\right)}$
So, the result is: $- \frac{i x \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)}}{2 \Gamma\left(\frac{4}{3}\right)}$
Now simplify:

$x \left(\operatorname{atan}{\left(\sqrt{x^{3} - 1} \right)} + \frac{3 i {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)}}{2}\right)$
Add the constant of integration:

$x \left(\operatorname{atan}{\left(\sqrt{x^{3} - 1} \right)} + \frac{3 i {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)}}{2}\right)+ \mathrm{constant}$

The answer is:

$x \left(\operatorname{atan}{\left(\sqrt{x^{3} - 1} \right)} + \frac{3 i {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)}}{2}\right)+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                  _                 
 |                                                                  |_  /1/3, 1/2 |  3\
 |     /   ________\                /   ________\   I*x*Gamma(1/3)* |   |         | x |
 |     |  /  3     |                |  /  3     |                  2  1 \  4/3    |   /
 | atan\\/  x  - 1 / dx = C + x*atan\\/  x  - 1 / + -----------------------------------
 |                                                              2*Gamma(4/3)           
/

$$\int \operatorname{atan}{\left(\sqrt{x^{3} - 1} \right)}\, dx = C + x \operatorname{atan}{\left(\sqrt{x^{3} - 1} \right)} + \frac{i x \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)}}{2 \Gamma\left(\frac{4}{3}\right)}$$

Simplify

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

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Integral of arctan(sqrt(x^3-1)) dx

Limits of integration:

The graph:

Piecewise:

The solution