Integral of atanx dx

The solution

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 |  atan(x) dx
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0

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

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

Detail solution

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 $v{\left(x \right)}$ :
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{x}{x^{2} + 1}\, dx = \frac{\int \frac{2 x}{x^{2} + 1}\, dx}{2}$
1. Let $u = x^{2} + 1$ .
  
  Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{1}{2 u}\, du$
  1. 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: $\frac{\log{\left(x^{2} + 1 \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$$x\,\arctan x-{{\log \left(x^2+1\right)}\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  log(2)   pi
- ------ + --
    2      4

$$-{{2\,\log 2-\pi}\over{4}}$$

  log(2)   pi
- ------ + --
    2      4

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

Simplify

Numerical answer [src]

0.438824573117476

0.438824573117476

The graph

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

Other calculators

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Identical expressions

Similar expressions

Integral of atanx dx

Limits of integration:

The graph:

Piecewise:

The solution