I'm trying to solve \(\displaystyle{\int_{{-\infty}}^{{\infty}}}{\frac{{{1}}}{{{\left({4}+{x}^{{2}}\right)}\sqrt{{{4}+{x}^{{2}}}}}}}{\left.{d}{x}\right.}\)

Alessandra Carrillo 2022-04-08 Answered
I'm trying to solve
1(4+x2)4+x2dx
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Answers (1)

oanhtih6
Answered 2022-04-09 Author has 10 answers
Therefore, after substitution, the integral becomes
π2π211+tan2t,dt
To proceed further, you need to use the following cousin of the Pythagorean theorem:
1+tan2t=sec2t
This identity is extremely important and useful in practice -- not the least in manipulating integrals like this. One should be reminded of this identity whenever one comes across an expression like 1+tan2t or 1+x2. By the way, the proof of this identity is based on the standard Pythagorean theorem:
1+tan2t=1+sin2tcos2t=cos2t+sin2tcos2t=1cos2t=sec2t
From this, it follows that
11+tan2t=|cost|
Thus the integral becomes
π2π2|cost|,dt
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