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Solving integrals/ 1\2*log(1+x^2)

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

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Piecewise:

The solution

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∫01log⁡(x2+1)2 dx\int\limits_{0}^{1} \frac{\log{\left(x^{2} + 1 \right)}}{2}\, dx 
Integral(log(1 + x^2)/2, (x, 0, 1))
Detail solution

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

    ∫log⁡(x2+1)2 dx=∫log⁡(x2+1) dx2\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:

      ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

      Let u(x)=log⁡(x2+1)u{\left(x \right)} = \log{\left(x^{2} + 1 \right)} and let dv⁡(x)=1\operatorname{dv}{\left(x \right)} = 1.

      Then du⁡(x)=2xx2+1\operatorname{du}{\left(x \right)} = \frac{2 x}{x^{2} + 1}.

      To find v(x)v{\left(x \right)}:

      1. The integral of a constant is the constant times the variable of integration:

        ∫1 dx=x\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:

      ∫2x2x2+1 dx=2∫x2x2+1 dx\int \frac{2 x^{2}}{x^{2} + 1}\, dx = 2 \int \frac{x^{2}}{x^{2} + 1}\, dx

      1. Rewrite the integrand:

        x2x2+1=1−1x2+1\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:

          ∫1 dx=x\int 1\, dx = x

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

          ∫(−1x2+1) dx=−∫1x2+1 dx\int \left(- \frac{1}{x^{2} + 1}\right)\, dx = - \int \frac{1}{x^{2} + 1}\, dx

            PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), True), (ArccothRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False), (ArctanhRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False)], context=1/(x**2 + 1), symbol=x)

          So, the result is: −atan⁡(x)- \operatorname{atan}{\left(x \right)}

        The result is: x−atan⁡(x)x - \operatorname{atan}{\left(x \right)}

      So, the result is: 2x−2atan⁡(x)2 x - 2 \operatorname{atan}{\left(x \right)}

    So, the result is: xlog⁡(x2+1)2−x+atan⁡(x)\frac{x \log{\left(x^{2} + 1 \right)}}{2} - x + \operatorname{atan}{\left(x \right)}

  2. Add the constant of integration:

    xlog⁡(x2+1)2−x+atan⁡(x)+constant\frac{x \log{\left(x^{2} + 1 \right)}}{2} - x + \operatorname{atan}{\left(x \right)}+ \mathrm{constant}

The answer is:

xlog⁡(x2+1)2−x+atan⁡(x)+constant\frac{x \log{\left(x^{2} + 1 \right)}}{2} - x + \operatorname{atan}{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                
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 |    /     2\                   /     2\          
 | log\1 + x /              x*log\1 + x /          
 | ----------- dx = C - x + ------------- + atan(x)
 |      2                         2                
 |                                                 
/
∫log⁡(x2+1)2 dx=C+xlog⁡(x2+1)2−x+atan⁡(x)\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
0.001.000.100.200.300.400.500.600.700.800.900.00.5
Plot the graph f(x)
The answer [src] 
     log(2)   pi
-1 + ------ + --
       2      4
−1+log⁡(2)2+π4-1 + \frac{\log{\left(2 \right)}}{2} + \frac{\pi}{4} 
=
= 
     log(2)   pi
-1 + ------ + --
       2      4
−1+log⁡(2)2+π4-1 + \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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