Integral of arctan(t) dt

The solution

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  t           
  /           
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 |  atan(t) dt
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0

$$\int\limits_{0}^{t} \operatorname{atan}{\left(t \right)}\, dt$$

Integral(atan(t), (t, 0, t))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(t \right)} = \operatorname{atan}{\left(t \right)}$ and let $\operatorname{dv}{\left(t \right)} = 1$ .

Then $\operatorname{du}{\left(t \right)} = \frac{1}{t^{2} + 1}$ .

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

$\int \frac{t}{t^{2} + 1}\, dt = \frac{\int \frac{2 t}{t^{2} + 1}\, dt}{2}$
1. Let $u = t^{2} + 1$ .
  
  Then let $du = 2 t dt$ 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:
  
  $\log{\left(t^{2} + 1 \right)}$
So, the result is: $\frac{\log{\left(t^{2} + 1 \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                    /     2\            
 |                  log\1 + t /            
 | atan(t) dt = C - ----------- + t*atan(t)
 |                       2                 
/

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

Simplify

The answer [src]

     /     2\            
  log\1 + t /            
- ----------- + t*atan(t)
       2

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

     /     2\            
  log\1 + t /            
- ----------- + t*atan(t)
       2

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

-log(1 + t^2)/2 + t*atan(t)

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

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Integral of arctan(t) dt

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The solution