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

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

The solution

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  1                       
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 |  (3 - 2*x)*atan(x)*1 dx
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0

$$\int\limits_{0}^{1} \left(3 - 2 x\right) \operatorname{atan}{\left(x \right)} 1\, dx$$

Integral(3 - 2*x*atan(x)*1, (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(3 - 2 x\right) \operatorname{atan}{\left(x \right)} 1 = - 2 x \operatorname{atan}{\left(x \right)} + 3 \operatorname{atan}{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 x \operatorname{atan}{\left(x \right)}\right)\, dx = - 2 \int x \operatorname{atan}{\left(x \right)}\, dx$
    1. Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = \operatorname{atan}{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = x$ .
      
      Then $\operatorname{du}{\left(x \right)} = \frac{1}{x^{2} + 1}$ .
      
      To find $v{\left(x \right)}$ :
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{x^{2}}{2 \left(x^{2} + 1\right)}\, dx = \frac{\int \frac{x^{2}}{x^{2} + 1}\, dx}{2}$
      
      Rewrite the integrand:
      
      $\frac{x^{2}}{x^{2} + 1} = 1 - \frac{1}{x^{2} + 1}$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x^{2} + 1}\right)\, dx = - \int \frac{1}{x^{2} + 1}\, dx$
      
      The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$ .
      
      So, the result is: $- \operatorname{atan}{\left(x \right)}$
      
      The result is: $x - \operatorname{atan}{\left(x \right)}$
      
      So, the result is: $\frac{x}{2} - \frac{\operatorname{atan}{\left(x \right)}}{2}$
    So, the result is: $- x^{2} \operatorname{atan}{\left(x \right)} + x - \operatorname{atan}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 \operatorname{atan}{\left(x \right)}\, dx = 3 \int \operatorname{atan}{\left(x \right)}\, dx$
    1. Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = \operatorname{atan}{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .
      
      Then $\operatorname{du}{\left(x \right)} = \frac{1}{x^{2} + 1}$ .
      
      To find $v{\left(x \right)}$ :
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{x}{x^{2} + 1}\, dx = \frac{\int \frac{2 x}{x^{2} + 1}\, dx}{2}$
      
      Let $u = x^{2} + 1$ .
      
      Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{2 u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x^{2} + 1 \right)}$
      
      So, the result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2}$
    So, the result is: $3 x \operatorname{atan}{\left(x \right)} - \frac{3 \log{\left(x^{2} + 1 \right)}}{2}$
  The result is: $- x^{2} \operatorname{atan}{\left(x \right)} + 3 x \operatorname{atan}{\left(x \right)} + x - \frac{3 \log{\left(x^{2} + 1 \right)}}{2} - \operatorname{atan}{\left(x \right)}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = \operatorname{atan}{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = 3 - 2 x$ .
  
  Then $\operatorname{du}{\left(x \right)} = \frac{1}{x^{2} + 1}$ .
  
  To find $v{\left(x \right)}$ :
  1. Integrate term-by-term:
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 3\, dx = 3 x$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 2 x\right)\, dx = - 2 \int x\, dx$
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
      
      So, the result is: $- x^{2}$
    The result is: $- x^{2} + 3 x$
  Now evaluate the sub-integral.
2. Rewrite the integrand:
  
  $\frac{- x^{2} + 3 x}{x^{2} + 1} = \frac{3 x + 1}{x^{2} + 1} - 1$
3. Integrate term-by-term:
  1. Rewrite the integrand:
    
    $\frac{3 x + 1}{x^{2} + 1} = \frac{3 x}{x^{2} + 1} + \frac{1}{x^{2} + 1}$
  2. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{3 x}{x^{2} + 1}\, dx = \frac{3 \int \frac{2 x}{x^{2} + 1}\, dx}{2}$
      
      Let $u = x^{2} + 1$ .
      
      Then let $du = 2 x dx$ and substitute $\frac{3 du}{2}$ :
      
      $\int \frac{3}{2 u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x^{2} + 1 \right)}$
      
      So, the result is: $\frac{3 \log{\left(x^{2} + 1 \right)}}{2}$
    1. The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$ .
    The result is: $\frac{3 \log{\left(x^{2} + 1 \right)}}{2} + \operatorname{atan}{\left(x \right)}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(-1\right)\, dx = - x$
  The result is: $- x + \frac{3 \log{\left(x^{2} + 1 \right)}}{2} + \operatorname{atan}{\left(x \right)}$
Method #3
1. Rewrite the integrand:
  
  $\left(3 - 2 x\right) \operatorname{atan}{\left(x \right)} 1 = - 2 x \operatorname{atan}{\left(x \right)} + 3 \operatorname{atan}{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 x \operatorname{atan}{\left(x \right)}\right)\, dx = - 2 \int x \operatorname{atan}{\left(x \right)}\, dx$
    1. Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = \operatorname{atan}{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = x$ .
      
      Then $\operatorname{du}{\left(x \right)} = \frac{1}{x^{2} + 1}$ .
      
      To find $v{\left(x \right)}$ :
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{x^{2}}{2 \left(x^{2} + 1\right)}\, dx = \frac{\int \frac{x^{2}}{x^{2} + 1}\, dx}{2}$
      
      Rewrite the integrand:
      
      $\frac{x^{2}}{x^{2} + 1} = 1 - \frac{1}{x^{2} + 1}$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x^{2} + 1}\right)\, dx = - \int \frac{1}{x^{2} + 1}\, dx$
      
      The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$ .
      
      So, the result is: $- \operatorname{atan}{\left(x \right)}$
      
      The result is: $x - \operatorname{atan}{\left(x \right)}$
      
      So, the result is: $\frac{x}{2} - \frac{\operatorname{atan}{\left(x \right)}}{2}$
    So, the result is: $- x^{2} \operatorname{atan}{\left(x \right)} + x - \operatorname{atan}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 \operatorname{atan}{\left(x \right)}\, dx = 3 \int \operatorname{atan}{\left(x \right)}\, dx$
    1. Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = \operatorname{atan}{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .
      
      Then $\operatorname{du}{\left(x \right)} = \frac{1}{x^{2} + 1}$ .
      
      To find $v{\left(x \right)}$ :
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{x}{x^{2} + 1}\, dx = \frac{\int \frac{2 x}{x^{2} + 1}\, dx}{2}$
      
      Let $u = x^{2} + 1$ .
      
      Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{2 u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x^{2} + 1 \right)}$
      
      So, the result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2}$
    So, the result is: $3 x \operatorname{atan}{\left(x \right)} - \frac{3 \log{\left(x^{2} + 1 \right)}}{2}$
  The result is: $- x^{2} \operatorname{atan}{\left(x \right)} + 3 x \operatorname{atan}{\left(x \right)} + x - \frac{3 \log{\left(x^{2} + 1 \right)}}{2} - \operatorname{atan}{\left(x \right)}$
Add the constant of integration:

$- x^{2} \operatorname{atan}{\left(x \right)} + 3 x \operatorname{atan}{\left(x \right)} + x - \frac{3 \log{\left(x^{2} + 1 \right)}}{2} - \operatorname{atan}{\left(x \right)}+ \mathrm{constant}$

The answer is:

$- x^{2} \operatorname{atan}{\left(x \right)} + 3 x \operatorname{atan}{\left(x \right)} + x - \frac{3 \log{\left(x^{2} + 1 \right)}}{2} - \operatorname{atan}{\left(x \right)}+ \mathrm{constant}$

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                                 
/

$$\int \left(3 - 2 x\right) \operatorname{atan}{\left(x \right)} 1\, dx = C - x^{2} \operatorname{atan}{\left(x \right)} + 3 x \operatorname{atan}{\left(x \right)} + x - \frac{3 \log{\left(x^{2} + 1 \right)}}{2} - \operatorname{atan}{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

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

$$- \frac{3 \log{\left(2 \right)}}{2} + \frac{\pi}{4} + 1$$

Упростить

Numerical answer [src]

0.74567739255753

0.74567739255753

The graph

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3