Integral of 13xarctgxdx dx

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\int\limits_{0}^{1} 13 x \operatorname{atan}{\left(x \right)} 1\, dx

Integral(13*x*atan(x)*1, (x, 0, 1))

Detail solution

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

$\int 13 x \operatorname{atan}{\left(x \right)} 1\, dx = 13 \int x \operatorname{atan}{\left(x \right)}\, dx$
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 $v{\left(x \right)}$ :
  1. 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:
  
  $\int \frac{x^{2}}{2 \left(x^{2} + 1\right)}\, dx = \frac{\int \frac{x^{2}}{x^{2} + 1}\, dx}{2}$
  1. Rewrite the integrand:
    
    $\frac{x^{2}}{x^{2} + 1} = 1 - \frac{1}{x^{2} + 1}$
  2. Integrate term-by-term:
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x^{2} + 1}\right)\, dx = - \int \frac{1}{x^{2} + 1}\, dx$
      1. 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: $\frac{13 x^{2} \operatorname{atan}{\left(x \right)}}{2} - \frac{13 x}{2} + \frac{13 \operatorname{atan}{\left(x \right)}}{2}$
Add the constant of integration:

$\frac{13 x^{2} \operatorname{atan}{\left(x \right)}}{2} - \frac{13 x}{2} + \frac{13 \operatorname{atan}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer is:

\frac{13 x^{2} \operatorname{atan}{\left(x \right)}}{2} - \frac{13 x}{2} + \frac{13 \operatorname{atan}{\left(x \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                                                2        
 |                         13*x   13*atan(x)   13*x *atan(x)
 | 13*x*atan(x)*1 dx = C - ---- + ---------- + -------------
 |                          2         2              2      
/

\int 13 x \operatorname{atan}{\left(x \right)} 1\, dx = C + \frac{13 x^{2} \operatorname{atan}{\left(x \right)}}{2} - \frac{13 x}{2} + \frac{13 \operatorname{atan}{\left(x \right)}}{2}

Simplify

The graph

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The answer [src]

  13   13*pi
- -- + -----
  2      4

- \frac{13}{2} + \frac{13 \pi}{4}

=

  13   13*pi
- -- + -----
  2      4

- \frac{13}{2} + \frac{13 \pi}{4}

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Numerical answer [src]

3.71017612416683

3.71017612416683

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Integral of 13xarctgxdx dx

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