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Integral of arctg2x-(3/(x^4))arctg2x dx

Limits of integration:

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Piecewise:

The solution

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  1                              
  /                              
 |                               
 |  /            3           \   
 |  |atan(2*x) - --*atan(2*x)| dx
 |  |             4          |   
 |  \            x           /   
 |                               
/                                
0                                
$$\int\limits_{0}^{1} \left(- \frac{3}{x^{4}} \operatorname{atan}{\left(2 x \right)} + \operatorname{atan}{\left(2 x \right)}\right)\, dx$$
Integral(atan(2*x) - 3/x^4*atan(2*x), (x, 0, 1))
Detail solution
  1. Integrate term-by-term:

    1. 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. 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. There are multiple ways to do this integral.

            Method #1

            1. Let .

              Then let and substitute :

              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 a constant times a function is the constant times the integral of the function:

                      1. The integral of is .

                      So, the result is:

                    Now substitute back in:

                  So, the result is:

                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:

                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 substitute back in:

            Method #2

            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. 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:

              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:

              1. The integral of is when :

              The result is:

            Method #3

            1. Rewrite the integrand:

            2. Rewrite the integrand:

            3. Integrate term-by-term:

              1. 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:

              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:

              1. The integral of is when :

              The result is:

            Method #4

            1. Rewrite the integrand:

            2. Rewrite the integrand:

            3. Integrate term-by-term:

              1. 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:

              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:

              1. The integral of is when :

              The result is:

          So, the result is:

        So, the result is:

      So, the result is:

    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:

    The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                                               
 |                                                            /       2\                          
 | /            3           \          1         / 2\   17*log\1 + 4*x /                 atan(2*x)
 | |atan(2*x) - --*atan(2*x)| dx = C + -- + 4*log\x / - ---------------- + x*atan(2*x) + ---------
 | |             4          |           2                      4                              3   
 | \            x           /          x                                                     x    
 |                                                                                                
/                                                                                                 
$$\int \left(- \frac{3}{x^{4}} \operatorname{atan}{\left(2 x \right)} + \operatorname{atan}{\left(2 x \right)}\right)\, dx = C + x \operatorname{atan}{\left(2 x \right)} + 4 \log{\left(x^{2} \right)} - \frac{17 \log{\left(4 x^{2} + 1 \right)}}{4} + \frac{1}{x^{2}} + \frac{\operatorname{atan}{\left(2 x \right)}}{x^{3}}$$
The answer [src]
-oo
$$-\infty$$
=
=
-oo
$$-\infty$$
-oo
Numerical answer [src]
-5.49219022742095e+38
-5.49219022742095e+38

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