Mister Exam

Integral of arctan(2x) dx

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The solution

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01atan(2x)dx\int\limits_{0}^{1} \operatorname{atan}{\left(2 x \right)}\, dx
Integral(atan(2*x), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=2xu = 2 x.

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

      atan(u)4du\int \frac{\operatorname{atan}{\left(u \right)}}{4}\, du

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

        atan(u)2du=atan(u)du2\int \frac{\operatorname{atan}{\left(u \right)}}{2}\, du = \frac{\int \operatorname{atan}{\left(u \right)}\, du}{2}

        1. Use integration by parts:

          udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

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

          Then du(u)=1u2+1\operatorname{du}{\left(u \right)} = \frac{1}{u^{2} + 1}.

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

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

            1du=u\int 1\, du = u

          Now evaluate the sub-integral.

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

          uu2+1du=2uu2+1du2\int \frac{u}{u^{2} + 1}\, du = \frac{\int \frac{2 u}{u^{2} + 1}\, du}{2}

          1. Let u=u2+1u = u^{2} + 1.

            Then let du=2ududu = 2 u du and substitute du2\frac{du}{2}:

            12udu\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(u2+1)\log{\left(u^{2} + 1 \right)}

          So, the result is: log(u2+1)2\frac{\log{\left(u^{2} + 1 \right)}}{2}

        So, the result is: uatan(u)2log(u2+1)4\frac{u \operatorname{atan}{\left(u \right)}}{2} - \frac{\log{\left(u^{2} + 1 \right)}}{4}

      Now substitute uu back in:

      xatan(2x)log(4x2+1)4x \operatorname{atan}{\left(2 x \right)} - \frac{\log{\left(4 x^{2} + 1 \right)}}{4}

    Method #2

    1. Use integration by parts:

      udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

      Let u(x)=atan(2x)u{\left(x \right)} = \operatorname{atan}{\left(2 x \right)} and let dv(x)=1\operatorname{dv}{\left(x \right)} = 1.

      Then du(x)=24x2+1\operatorname{du}{\left(x \right)} = \frac{2}{4 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:

        1dx=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:

      2x4x2+1dx=2x4x2+1dx\int \frac{2 x}{4 x^{2} + 1}\, dx = 2 \int \frac{x}{4 x^{2} + 1}\, dx

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

        x4x2+1dx=8x4x2+1dx8\int \frac{x}{4 x^{2} + 1}\, dx = \frac{\int \frac{8 x}{4 x^{2} + 1}\, dx}{8}

        1. Let u=4x2+1u = 4 x^{2} + 1.

          Then let du=8xdxdu = 8 x dx and substitute du8\frac{du}{8}:

          18udu\int \frac{1}{8 u}\, du

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

          Now substitute uu back in:

          log(4x2+1)\log{\left(4 x^{2} + 1 \right)}

        So, the result is: log(4x2+1)8\frac{\log{\left(4 x^{2} + 1 \right)}}{8}

      So, the result is: log(4x2+1)4\frac{\log{\left(4 x^{2} + 1 \right)}}{4}

  2. Add the constant of integration:

    xatan(2x)log(4x2+1)4+constantx \operatorname{atan}{\left(2 x \right)} - \frac{\log{\left(4 x^{2} + 1 \right)}}{4}+ \mathrm{constant}


The answer is:

xatan(2x)log(4x2+1)4+constantx \operatorname{atan}{\left(2 x \right)} - \frac{\log{\left(4 x^{2} + 1 \right)}}{4}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                      /       2\              
 |                    log\1 + 4*x /              
 | atan(2*x) dx = C - ------------- + x*atan(2*x)
 |                          4                    
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atan(2x)dx=C+xatan(2x)log(4x2+1)4\int \operatorname{atan}{\left(2 x \right)}\, dx = C + x \operatorname{atan}{\left(2 x \right)} - \frac{\log{\left(4 x^{2} + 1 \right)}}{4}
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
The answer [src]
  log(5)          
- ------ + atan(2)
    4             
log(5)4+atan(2)- \frac{\log{\left(5 \right)}}{4} + \operatorname{atan}{\left(2 \right)}
=
=
  log(5)          
- ------ + atan(2)
    4             
log(5)4+atan(2)- \frac{\log{\left(5 \right)}}{4} + \operatorname{atan}{\left(2 \right)}
Numerical answer [src]
0.704789239685565
0.704789239685565
The graph
Integral of arctan(2x) dx

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