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Integral of 1\2*log(1+x^2)
Identical expressions
one \ two *log(one +x^ two)
1\2 multiply by logarithm of (1 plus x squared )
one \ two multiply by logarithm of (one plus x to the power of two)
1\2*log(1+x2)
1\2*log1+x2
1\2*log(1+x²)
1\2*log(1+x to the power of 2)
1\2log(1+x^2)
1\2log(1+x2)
1\2log1+x2
1\2log1+x^2
1\2*log(1+x^2)dx
Similar expressions
1\2*log(1-x^2)

Integral of 1\2*log(1+x^2) dx

The solution

You have entered [src]

  1               
  /               
 |                
 |     /     2\   
 |  log\1 + x /   
 |  ----------- dx
 |       2        
 |                
/                 
0

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

Integral(log(1 + x^2)/2, (x, 0, 1))

Detail solution

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

$\int \frac{\log{\left(x^{2} + 1 \right)}}{2}\, dx = \frac{\int \log{\left(x^{2} + 1 \right)}\, dx}{2}$
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = \log{\left(x^{2} + 1 \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .
  
  Then $\operatorname{du}{\left(x \right)} = \frac{2 x}{x^{2} + 1}$ .
  
  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.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 x^{2}}{x^{2} + 1}\, dx = 2 \int \frac{x^{2}}{x^{2} + 1}\, dx$
  1. Rewrite the integrand:
    
    $\frac{x^{2}}{x^{2} + 1} = 1 - \frac{1}{x^{2} + 1}$
  2. Integrate term-by-term:
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x^{2} + 1}\right)\, dx = - \int \frac{1}{x^{2} + 1}\, dx$
      So, the result is: $- \operatorname{atan}{\left(x \right)}$
    The result is: $x - \operatorname{atan}{\left(x \right)}$
  So, the result is: $2 x - 2 \operatorname{atan}{\left(x \right)}$
So, the result is: $\frac{x \log{\left(x^{2} + 1 \right)}}{2} - x + \operatorname{atan}{\left(x \right)}$
Add the constant of integration:

$\frac{x \log{\left(x^{2} + 1 \right)}}{2} - x + \operatorname{atan}{\left(x \right)}+ \mathrm{constant}$

The answer is:

$\frac{x \log{\left(x^{2} + 1 \right)}}{2} - x + \operatorname{atan}{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

     log(2)   pi
-1 + ------ + --
       2      4

$$-1 + \frac{\log{\left(2 \right)}}{2} + \frac{\pi}{4}$$

     log(2)   pi
-1 + ------ + --
       2      4

$$-1 + \frac{\log{\left(2 \right)}}{2} + \frac{\pi}{4}$$

-1 + log(2)/2 + pi/4

Numerical answer [src]

0.131971753677421

0.131971753677421

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

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Integral of 1\2*log(1+x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution