Given:
#f(x) = 3/(7x^2+4)#
Evaluate first:
#lim_(t->oo) int_0^t 3/(7x^2+4) dx = lim_(t->oo) 3/4 int_0^t dx/(7/4x^2+1) #
#lim_(t->oo) int_0^t 3/(7x^2+4) dx = lim_(t->oo) 3/(2sqrt7) int_0^t (d(sqrt7/2x))/( (sqrt7/2x)^2+1) #
#lim_(t->oo) int_0^t 3/(7x^2+4) dx = lim_(t->oo) 3/(2sqrt7) [arctan(sqrt7/2x)]_0^t #
#lim_(t->oo) int_0^t 3/(7x^2+4) dx = lim_(t->oo) 3/(2sqrt7) arctan(sqrt7/2t) #
#lim_(t->oo) int_0^t 3/(7x^2+4) dx = (3pi)/(4sqrt7)#
Then the integral:
#int_0^oo 3/(7x^2+4)dx = (3pi)/(4sqrt7)#
is convergent.
Evaluate then:
#int_(-t)^0 3/(7x^2+4)dx#
substitute now: #u=-x#:
#int_(-t)^0 3/(7x^2+4)dx = - int_t^0 3/(7(-u)^2+4)du = int_0^t 3/(7u^2+4)du#
so that also:
#int_(-oo)^0 3/(7x^2+4)dx = (3pi)/(4sqrt7)#
is convergent.
Finally then:
#int_(-oo)^(+oo) 3/(7x^2+4)dx = (3pi)/(2sqrt7)#