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(3-2x)*arctg(x)dx

Integral of (3-2x)*arctg(x)dx dx

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The solution

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  1                       
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 |  (3 - 2*x)*atan(x)*1 dx
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$$\int\limits_{0}^{1} \left(3 - 2 x\right) \operatorname{atan}{\left(x \right)} 1\, dx$$
Integral(3 - 2*x*atan(x)*1, (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      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 is when :

          Now evaluate the sub-integral.

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

          1. Rewrite the integrand:

          2. Integrate term-by-term:

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

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

              1. The integral of is .

              So, the result is:

            The result is:

          So, the result is:

        So, the result is:

      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:

      The result is:

    Method #2

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. Integrate term-by-term:

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

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

          1. The integral of is when :

          So, the result is:

        The result is:

      Now evaluate the sub-integral.

    2. Rewrite the integrand:

    3. Integrate term-by-term:

      1. Rewrite the integrand:

      2. Integrate term-by-term:

        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:

        1. The integral of is .

        The result is:

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

      The result is:

    Method #3

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      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 is when :

          Now evaluate the sub-integral.

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

          1. Rewrite the integrand:

          2. Integrate term-by-term:

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

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

              1. The integral of is .

              So, the result is:

            The result is:

          So, the result is:

        So, the result is:

      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:

      The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                /     2\                           
 |                                            3*log\1 + x /    2                      
 | (3 - 2*x)*atan(x)*1 dx = C + x - atan(x) - ------------- - x *atan(x) + 3*x*atan(x)
 |                                                  2                                 
/                                                                                     
$$\int \left(3 - 2 x\right) \operatorname{atan}{\left(x \right)} 1\, dx = C - x^{2} \operatorname{atan}{\left(x \right)} + 3 x \operatorname{atan}{\left(x \right)} + x - \frac{3 \log{\left(x^{2} + 1 \right)}}{2} - \operatorname{atan}{\left(x \right)}$$
The graph
The answer [src]
    3*log(2)   pi
1 - -------- + --
       2       4 
$$- \frac{3 \log{\left(2 \right)}}{2} + \frac{\pi}{4} + 1$$
=
=
    3*log(2)   pi
1 - -------- + --
       2       4 
$$- \frac{3 \log{\left(2 \right)}}{2} + \frac{\pi}{4} + 1$$
Numerical answer [src]
0.74567739255753
0.74567739255753
The graph
Integral of (3-2x)*arctg(x)dx dx

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