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(2x-x^2)arctgx/2dx

Integral of (2x+x^2)arctgx/2dx dx

The solution

You have entered [src]

  1                      
  /                      
 |                       
 |  /       2\           
 |  \2*x + x /*acot(x)   
 |  ------------------ dx
 |          2            
 |                       
/                        
0

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

Integral(((2*x + x^2)*acot(x))/2, (x, 0, 1))

Detail solution

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

$\int \frac{\left(x^{2} + 2 x\right) \operatorname{acot}{\left(x \right)}}{2}\, dx = \frac{\int \left(x^{2} + 2 x\right) \operatorname{acot}{\left(x \right)}\, dx}{2}$
1. Rewrite the integrand:
  
  $\left(x^{2} + 2 x\right) \operatorname{acot}{\left(x \right)} = x^{2} \operatorname{acot}{\left(x \right)} + 2 x \operatorname{acot}{\left(x \right)}$
2. Integrate term-by-term:
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\frac{x^{3} \operatorname{acot}{\left(x \right)}}{3} + \frac{x^{2}}{6} - \frac{\log{\left(x^{2} + 1 \right)}}{6}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 x \operatorname{acot}{\left(x \right)}\, dx = 2 \int x \operatorname{acot}{\left(x \right)}\, dx$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\frac{x^{2} \operatorname{acot}{\left(x \right)}}{2} + \frac{x}{2} + \frac{\operatorname{acot}{\left(x \right)}}{2}$
    So, the result is: $x^{2} \operatorname{acot}{\left(x \right)} + x + \operatorname{acot}{\left(x \right)}$
  The result is: $\frac{x^{3} \operatorname{acot}{\left(x \right)}}{3} + x^{2} \operatorname{acot}{\left(x \right)} + \frac{x^{2}}{6} + x - \frac{\log{\left(x^{2} + 1 \right)}}{6} + \operatorname{acot}{\left(x \right)}$
So, the result is: $\frac{x^{3} \operatorname{acot}{\left(x \right)}}{6} + \frac{x^{2} \operatorname{acot}{\left(x \right)}}{2} + \frac{x^{2}}{12} + \frac{x}{2} - \frac{\log{\left(x^{2} + 1 \right)}}{12} + \frac{\operatorname{acot}{\left(x \right)}}{2}$
Add the constant of integration:

$\frac{x^{3} \operatorname{acot}{\left(x \right)}}{6} + \frac{x^{2} \operatorname{acot}{\left(x \right)}}{2} + \frac{x^{2}}{12} + \frac{x}{2} - \frac{\log{\left(x^{2} + 1 \right)}}{12} + \frac{\operatorname{acot}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{x^{3} \operatorname{acot}{\left(x \right)}}{6} + \frac{x^{2} \operatorname{acot}{\left(x \right)}}{2} + \frac{x^{2}}{12} + \frac{x}{2} - \frac{\log{\left(x^{2} + 1 \right)}}{12} + \frac{\operatorname{acot}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                    
 |                                                                                     
 | /       2\                                   /     2\    2    2            3        
 | \2*x + x /*acot(x)          x   acot(x)   log\1 + x /   x    x *acot(x)   x *acot(x)
 | ------------------ dx = C + - + ------- - ----------- + -- + ---------- + ----------
 |         2                   2      2           12       12       2            6     
 |                                                                                     
/

$$\int \frac{\left(x^{2} + 2 x\right) \operatorname{acot}{\left(x \right)}}{2}\, dx = C + \frac{x^{3} \operatorname{acot}{\left(x \right)}}{6} + \frac{x^{2} \operatorname{acot}{\left(x \right)}}{2} + \frac{x^{2}}{12} + \frac{x}{2} - \frac{\log{\left(x^{2} + 1 \right)}}{12} + \frac{\operatorname{acot}{\left(x \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

7    log(2)   pi
-- - ------ + --
12     12     24

$$- \frac{\log{\left(2 \right)}}{12} + \frac{\pi}{24} + \frac{7}{12}$$

7    log(2)   pi
-- - ------ + --
12     12     24

$$- \frac{\log{\left(2 \right)}}{12} + \frac{\pi}{24} + \frac{7}{12}$$

7/12 - log(2)/12 + pi/24

Numerical answer [src]

0.656470762186246

0.656470762186246

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

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Integral of (2x+x^2)arctgx/2dx dx

Limits of integration:

The graph:

Piecewise:

The solution