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Solving integrals/ (1/sqrt(x))*arctg(x/2+x)

Integral of (1/sqrt(x))*arctg(x/2+x) dx

Limits of integration:

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Piecewise:

The solution

You have entered [src] 
 oo               
  /               
 |                
 |      /x    \   
 |  atan|- + x|   
 |      \2    /   
 |  ----------- dx
 |       ___      
 |     \/ x       
 |                
/                 
2
∫2∞atan⁡(x2+x)x dx\int\limits_{2}^{\infty} \frac{\operatorname{atan}{\left(\frac{x}{2} + x \right)}}{\sqrt{x}}\, dx 
Integral(atan(x/2 + x)/sqrt(x), (x, 2, oo))
The answer (Indefinite) [src] 
  /                                                                                                                                                                               
 |                                                                                                                      /            ___   ___\            /            ___   ___\
 |     /x    \                                                                                                   ___    |2       2*\/ 3 *\/ x |     ___    |2       2*\/ 3 *\/ x |
 | atan|- + x|                                    ___     /      ___   ___\       ___     /       ___   ___\   \/ 3 *log|- + x - -------------|   \/ 3 *log|- + x + -------------|
 |     \2    /              ___     /x    \   2*\/ 3 *atan\1 + \/ 3 *\/ x /   2*\/ 3 *atan\-1 + \/ 3 *\/ x /            \3             3      /            \3             3      /
 | ----------- dx = C + 2*\/ x *atan|- + x| - ----------------------------- - ------------------------------ - -------------------------------- + --------------------------------
 |      ___                         \2    /                 3                               3                                 3                                  3                
 |    \/ x                                                                                                                                                                        
 |                                                                                                                                                                                
/
∫atan⁡(x2+x)x dx=C+2xatan⁡(x2+x)−3log⁡(−23x3+x+23)3+3log⁡(23x3+x+23)3−23atan⁡(3x−1)3−23atan⁡(3x+1)3\int \frac{\operatorname{atan}{\left(\frac{x}{2} + x \right)}}{\sqrt{x}}\, dx = C + 2 \sqrt{x} \operatorname{atan}{\left(\frac{x}{2} + x \right)} - \frac{\sqrt{3} \log{\left(- \frac{2 \sqrt{3} \sqrt{x}}{3} + x + \frac{2}{3} \right)}}{3} + \frac{\sqrt{3} \log{\left(\frac{2 \sqrt{3} \sqrt{x}}{3} + x + \frac{2}{3} \right)}}{3} - \frac{2 \sqrt{3} \operatorname{atan}{\left(\sqrt{3} \sqrt{x} - 1 \right)}}{3} - \frac{2 \sqrt{3} \operatorname{atan}{\left(\sqrt{3} \sqrt{x} + 1 \right)}}{3}
Simplify

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

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