Mister Exam

Other calculators

(3-2x)*arctg(x)dx
Solving integrals/ (3-2x)*arctg(x)dx

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1                       
  /                       
 |                        
 |  (3 - 2*x)*atan(x)*1 dx
 |                        
/                         
0
∫01(3−2x)atan⁡(x)1 dx\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:

      (3−2x)atan⁡(x)1=−2xatan⁡(x)+3atan⁡(x)\left(3 - 2 x\right) \operatorname{atan}{\left(x \right)} 1 = - 2 x \operatorname{atan}{\left(x \right)} + 3 \operatorname{atan}{\left(x \right)}

    2. Integrate term-by-term:

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

        ∫(−2xatan⁡(x)) dx=−2∫xatan⁡(x) dx\int \left(- 2 x \operatorname{atan}{\left(x \right)}\right)\, dx = - 2 \int x \operatorname{atan}{\left(x \right)}\, dx

        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)=x\operatorname{dv}{\left(x \right)} = x.

          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 xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

          Now evaluate the sub-integral.

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

          ∫x22(x2+1) dx=∫x2x2+1 dx2\int \frac{x^{2}}{2 \left(x^{2} + 1\right)}\, dx = \frac{\int \frac{x^{2}}{x^{2} + 1}\, dx}{2}

          1. Rewrite the integrand:

            x2x2+1=1−1x2+1\frac{x^{2}}{x^{2} + 1} = 1 - \frac{1}{x^{2} + 1}

          2. Integrate term-by-term:

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

              ∫1 dx=x\int 1\, dx = x

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

              ∫(−1x2+1) dx=−∫1x2+1 dx\int \left(- \frac{1}{x^{2} + 1}\right)\, dx = - \int \frac{1}{x^{2} + 1}\, dx

              1. The integral of 1x2+1\frac{1}{x^{2} + 1} is atan⁡(x)\operatorname{atan}{\left(x \right)}.

              So, the result is: −atan⁡(x)- \operatorname{atan}{\left(x \right)}

            The result is: x−atan⁡(x)x - \operatorname{atan}{\left(x \right)}

          So, the result is: x2−atan⁡(x)2\frac{x}{2} - \frac{\operatorname{atan}{\left(x \right)}}{2}

        So, the result is: −x2atan⁡(x)+x−atan⁡(x)- x^{2} \operatorname{atan}{\left(x \right)} + x - \operatorname{atan}{\left(x \right)}

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

        ∫3atan⁡(x) dx=3∫atan⁡(x) dx\int 3 \operatorname{atan}{\left(x \right)}\, dx = 3 \int \operatorname{atan}{\left(x \right)}\, dx

        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}

        So, the result is: 3xatan⁡(x)−3log⁡(x2+1)23 x \operatorname{atan}{\left(x \right)} - \frac{3 \log{\left(x^{2} + 1 \right)}}{2}

      The result is: −x2atan⁡(x)+3xatan⁡(x)+x−3log⁡(x2+1)2−atan⁡(x)- 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)}

    Method #2

    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)=3−2x\operatorname{dv}{\left(x \right)} = 3 - 2 x.

      Then du⁡(x)=1x2+1\operatorname{du}{\left(x \right)} = \frac{1}{x^{2} + 1}.

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

      1. Integrate term-by-term:

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

          ∫3 dx=3x\int 3\, dx = 3 x

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

          ∫(−2x) dx=−2∫x dx\int \left(- 2 x\right)\, dx = - 2 \int x\, dx

          1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

          So, the result is: −x2- x^{2}

        The result is: −x2+3x- x^{2} + 3 x

      Now evaluate the sub-integral.

    2. Rewrite the integrand:

      −x2+3xx2+1=3x+1x2+1−1\frac{- x^{2} + 3 x}{x^{2} + 1} = \frac{3 x + 1}{x^{2} + 1} - 1

    3. Integrate term-by-term:

      1. Rewrite the integrand:

        3x+1x2+1=3xx2+1+1x2+1\frac{3 x + 1}{x^{2} + 1} = \frac{3 x}{x^{2} + 1} + \frac{1}{x^{2} + 1}

      2. Integrate term-by-term:

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

          ∫3xx2+1 dx=3∫2xx2+1 dx2\int \frac{3 x}{x^{2} + 1}\, dx = \frac{3 \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 3du2\frac{3 du}{2}:

            ∫32u du\int \frac{3}{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: 3log⁡(x2+1)2\frac{3 \log{\left(x^{2} + 1 \right)}}{2}

        1. The integral of 1x2+1\frac{1}{x^{2} + 1} is atan⁡(x)\operatorname{atan}{\left(x \right)}.

        The result is: 3log⁡(x2+1)2+atan⁡(x)\frac{3 \log{\left(x^{2} + 1 \right)}}{2} + \operatorname{atan}{\left(x \right)}

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

        ∫(−1) dx=−x\int \left(-1\right)\, dx = - x

      The result is: −x+3log⁡(x2+1)2+atan⁡(x)- x + \frac{3 \log{\left(x^{2} + 1 \right)}}{2} + \operatorname{atan}{\left(x \right)}

    Method #3

    1. Rewrite the integrand:

      (3−2x)atan⁡(x)1=−2xatan⁡(x)+3atan⁡(x)\left(3 - 2 x\right) \operatorname{atan}{\left(x \right)} 1 = - 2 x \operatorname{atan}{\left(x \right)} + 3 \operatorname{atan}{\left(x \right)}

    2. Integrate term-by-term:

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

        ∫(−2xatan⁡(x)) dx=−2∫xatan⁡(x) dx\int \left(- 2 x \operatorname{atan}{\left(x \right)}\right)\, dx = - 2 \int x \operatorname{atan}{\left(x \right)}\, dx

        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)=x\operatorname{dv}{\left(x \right)} = x.

          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 xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

          Now evaluate the sub-integral.

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

          ∫x22(x2+1) dx=∫x2x2+1 dx2\int \frac{x^{2}}{2 \left(x^{2} + 1\right)}\, dx = \frac{\int \frac{x^{2}}{x^{2} + 1}\, dx}{2}

          1. Rewrite the integrand:

            x2x2+1=1−1x2+1\frac{x^{2}}{x^{2} + 1} = 1 - \frac{1}{x^{2} + 1}

          2. Integrate term-by-term:

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

              ∫1 dx=x\int 1\, dx = x

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

              ∫(−1x2+1) dx=−∫1x2+1 dx\int \left(- \frac{1}{x^{2} + 1}\right)\, dx = - \int \frac{1}{x^{2} + 1}\, dx

              1. The integral of 1x2+1\frac{1}{x^{2} + 1} is atan⁡(x)\operatorname{atan}{\left(x \right)}.

              So, the result is: −atan⁡(x)- \operatorname{atan}{\left(x \right)}

            The result is: x−atan⁡(x)x - \operatorname{atan}{\left(x \right)}

          So, the result is: x2−atan⁡(x)2\frac{x}{2} - \frac{\operatorname{atan}{\left(x \right)}}{2}

        So, the result is: −x2atan⁡(x)+x−atan⁡(x)- x^{2} \operatorname{atan}{\left(x \right)} + x - \operatorname{atan}{\left(x \right)}

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

        ∫3atan⁡(x) dx=3∫atan⁡(x) dx\int 3 \operatorname{atan}{\left(x \right)}\, dx = 3 \int \operatorname{atan}{\left(x \right)}\, dx

        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}

        So, the result is: 3xatan⁡(x)−3log⁡(x2+1)23 x \operatorname{atan}{\left(x \right)} - \frac{3 \log{\left(x^{2} + 1 \right)}}{2}

      The result is: −x2atan⁡(x)+3xatan⁡(x)+x−3log⁡(x2+1)2−atan⁡(x)- 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)}

  2. Add the constant of integration:

    −x2atan⁡(x)+3xatan⁡(x)+x−3log⁡(x2+1)2−atan⁡(x)+constant- 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)}+ \mathrm{constant}

The answer is:

−x2atan⁡(x)+3xatan⁡(x)+x−3log⁡(x2+1)2−atan⁡(x)+constant- 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)}+ \mathrm{constant}

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                                 
/
∫(3−2x)atan⁡(x)1 dx=C−x2atan⁡(x)+3xatan⁡(x)+x−3log⁡(x2+1)2−atan⁡(x)\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)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9001
Plot the graph f(x)
The answer [src] 
    3*log(2)   pi
1 - -------- + --
       2       4
−3log⁡(2)2+π4+1- \frac{3 \log{\left(2 \right)}}{2} + \frac{\pi}{4} + 1 
=
= 
    3*log(2)   pi
1 - -------- + --
       2       4
−3log⁡(2)2+π4+1- \frac{3 \log{\left(2 \right)}}{2} + \frac{\pi}{4} + 1
Упростить
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.

    © Mister Exam – Calculator