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sqrt(arctan(x))/(1-x^2)

Solving integrals/ sqrt(arctan(x))/(1+x^2)

Integral of sqrt(arctan(x))/(1+x^2) dx

The solution

You have entered [src]

 pi               
 --               
 4                
  /               
 |                
 |    _________   
 |  \/ atan(x)    
 |  ----------- dx
 |          2     
 |     1 + x      
 |                
/                 
0

$$\int\limits_{0}^{\frac{\pi}{4}} \frac{\sqrt{\operatorname{atan}{\left(x \right)}}}{x^{2} + 1}\, dx$$

Integral(sqrt(atan(x))/(1 + x^2), (x, 0, pi/4))

Detail solution

Let $u = \operatorname{atan}{\left(x \right)}$ .

Then let $du = \frac{dx}{x^{2} + 1}$ and substitute $du$ :

$\int \sqrt{u}\, du$
1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}$
Now substitute $u$ back in:

$\frac{2 \operatorname{atan}^{\frac{3}{2}}{\left(x \right)}}{3}$
Add the constant of integration:

$\frac{2 \operatorname{atan}^{\frac{3}{2}}{\left(x \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{2 \operatorname{atan}^{\frac{3}{2}}{\left(x \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                                  
 |   _________                3/2   
 | \/ atan(x)           2*atan   (x)
 | ----------- dx = C + ------------
 |         2                 3      
 |    1 + x                         
 |                                  
/

$${{2\,\left(\arctan x\right)^{{{3}\over{2}}}}\over{3}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

      3/2/pi\
2*atan   |--|
         \4 /
-------------
      3

$$\frac{2 \operatorname{atan}^{\frac{3}{2}}{\left(\frac{\pi}{4} \right)}}{3}$$

      3/2/pi\
2*atan   |--|
         \4 /
-------------
      3

$$\frac{2 \operatorname{atan}^{\frac{3}{2}}{\left(\frac{\pi}{4} \right)}}{3}$$

Simplify

Numerical answer [src]

0.362158550095807

0.362158550095807

The graph

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

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Identical expressions

Similar expressions

Integral of sqrt(arctan(x))/(1+x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution