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Integral of arctg*(4*x) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

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 |  atan(4*x) dx
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$$\int\limits_{0}^{0} \operatorname{atan}{\left(4 x \right)}\, dx$$
Integral(atan(4*x), (x, 0, 0))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

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

        1. Use integration by parts:

          Let and let .

          Then .

          To find :

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

          Now evaluate the sub-integral.

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

          1. Let .

            Then let and substitute :

            1. The integral of is .

            Now substitute back in:

          So, the result is:

        So, the result is:

      Now substitute back in:

    Method #2

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

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

      Now evaluate the sub-integral.

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

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

        1. Let .

          Then let and substitute :

          1. The integral of is .

          Now substitute back in:

        So, the result is:

      So, the result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                      /        2\              
 |                    log\1 + 16*x /              
 | atan(4*x) dx = C - -------------- + x*atan(4*x)
 |                          8                     
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$$\int \operatorname{atan}{\left(4 x \right)}\, dx = C + x \operatorname{atan}{\left(4 x \right)} - \frac{\log{\left(16 x^{2} + 1 \right)}}{8}$$
The graph
The answer [src]
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Numerical answer [src]
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    Use the examples entering the upper and lower limits of integration.