Solve the trigonometric integral int(arctan(t))dt

 Exercise

$\int\arctan\left(t\right)dt$

 

Apply integration by parts: $u=arctan(t)$, $v'=1$

$t\arctan\left(t\right)-\int\frac{t}{1+t^2}dt$

 

The integral $-\int\frac{t}{1+t^2}dt$ results in: $-\frac{1}{2}\ln\left(1+t^2\right)$

$-\frac{1}{2}\ln\left(1+t^2\right)$

 

Gather the results of all integrals

$t\arctan\left(t\right)-\frac{1}{2}\ln\left|1+t^2\right|$

 

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$t\arctan\left(t\right)-\frac{1}{2}\ln\left|1+t^2\right|+C_0$

$t\arctan\left(t\right)-\frac{1}{2}\ln\left|1+t^2\right|+C_0$

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