A calculus problem by Jose Sacramento

Calculus Level 3

If $\displaystyle \int_{-\infty}^2 \dfrac1{4+x^2} \, dx = \dfrac ab\pi$ , where $a$ and $b$ are coprime positive integers, find $a+b$ .

The answer is 11.

2 solutions

Steven Chase
Sep 9, 2016

Henry Lembeck
Sep 10, 2016

Let x=2tan(a). Then dx=2sec^2(a) and a=arctan(x/2). The integrand, then, can be rewritten as 2sec^2(a)/(4+4tan^2(a)), which is equal to 2sec^2(a)/(4sec^2(a))=1/2. Integrating with respect to a we have 1/2a which is equal to 1/2arctan(x/2). Limit as x -> -infinity is -1/4pi and as x->2 is 1/8pi (limit as x-2 of arctan(x/2) is pi/4) so (1/8-(-1/4))pi =3/8pi. 3+8=11.

Sorry i tried to use latex but it wasnt appearing to work in preview so i stopped

A calculus problem by Jose Sacramento

The answer is 11.

2 solutions

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