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Solving integrals/ arctan

Integral of arctan dx

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The solution

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∫01atan⁡(x) dx\int\limits_{0}^{1} \operatorname{atan}{\left(x \right)}\, dx 
Integral(atan(x), (x, 0, 1))
Detail solution

  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)=atan⁡(x)u{\left(x \right)} = \operatorname{atan}{\left(x \right)} and let dv⁡(x)=1\operatorname{dv}{\left(x \right)} = 1.

    Then du⁡(x)=1x2+1\operatorname{du}{\left(x \right)} = \frac{1}{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:

    ∫xx2+1 dx=∫2xx2+1 dx2\int \frac{x}{x^{2} + 1}\, dx = \frac{\int \frac{2 x}{x^{2} + 1}\, dx}{2}

    1. Let u=x2+1u = x^{2} + 1.

      Then let du=2xdxdu = 2 x dx and substitute du2\frac{du}{2}:

      ∫12u du\int \frac{1}{2 u}\, du

      1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

      Now substitute uu back in:

      log⁡(x2+1)\log{\left(x^{2} + 1 \right)}

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
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 | atan(x) dx = C - ----------- + x*atan(x)
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∫atan⁡(x) dx=C+xatan⁡(x)−log⁡(x2+1)2\int \operatorname{atan}{\left(x \right)}\, dx = C + x \operatorname{atan}{\left(x \right)} - \frac{\log{\left(x^{2} + 1 \right)}}{2}
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] 
  log(2)   pi
- ------ + --
    2      4
−log⁡(2)2+π4- \frac{\log{\left(2 \right)}}{2} + \frac{\pi}{4} 
=
= 
  log(2)   pi
- ------ + --
    2      4
−log⁡(2)2+π4- \frac{\log{\left(2 \right)}}{2} + \frac{\pi}{4} 
-log(2)/2 + pi/4
Numerical answer [src] 
0.438824573117476
0.438824573117476
The graph
Integral of arctan dx

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

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