How do you integrate $( (arctan(x/2) ) ÷ (4+x²) dx$ ?

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Answer 1 · 2016-09-12T19:18:05

$1/4arctan^2(x/2)+C$

$intarctan(x/2)/(4+x^2)dx$

We can use substitution. Let $u=arctan(x/2)$ . Taking the derivative, we see that $du=(1/2)/(1+(x/2)^2)dx=(1/2)/((4+x^2)/4)dx=2/(4+x^2)dx$ .

Thus:

$=1/2int(2arctan(x/2))/(4+x^2)dx=1/2intudu=1/2(u^2/2)=u^2/4=1/4arctan^2(x/2)+C$

How do you integrate ( (arctan(x/2) ) ÷ (4+x²) dx( (arctan(x/2) ) ÷ (4+x²) dx?