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

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

Limits of integration:

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The graph:

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Piecewise:

The solution

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∫01x2x2+1 dx\int\limits_{0}^{1} \frac{x^{2}}{x^{2} + 1}\, dx 
Integral(x^2/(1 + x^2), (x, 0, 1))
Detail solution

  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)}

  3. Add the constant of integration:

    x−atan⁡(x)+constantx - \operatorname{atan}{\left(x \right)}+ \mathrm{constant}

The answer is:

x−atan⁡(x)+constantx - \operatorname{atan}{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
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∫x2x2+1 dx=C+x−atan⁡(x)\int \frac{x^{2}}{x^{2} + 1}\, dx = C + x - \operatorname{atan}{\left(x \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.01.0
Plot the graph f(x)
The answer [src] 
    pi
1 - --
    4
1−π41 - \frac{\pi}{4} 
=
= 
    pi
1 - --
    4
1−π41 - \frac{\pi}{4} 
1 - pi/4
Numerical answer [src] 
0.214601836602552
0.214601836602552
The graph
Integral of (x^2)/(1+x^2) dx

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

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