How would I use the Comparison Test in calculus to solve the integral #(cos(4x) +3) / (3x^3 + 1)# from 1 to infinity?

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Answer 1 · 2015-09-12T12:51:19

I'm not sure what you mean by "solve".

By comparison, we can see that the integral converges.

#(cos(4x) +3) / (3x^3 + 1) <= 4/(3x^3+1)# for #x >= 1#

graph{(y-100((cos (4x)+3)/(3x^3+1)))(y-((400)/(3x^3+1)))=0 [0.33, 11.433, -0.75, 4.8]}
(Both multiplied by 100 for clarity of relationship.)

And #4/(3x^3+1) <= 4/x^3# for #x >= 1#

So

#(cos(4x) +3) / (3x^3 + 1) <= 4/x^3# for #x >= 1#

graph{(y-100((cos (4x)+3)/(3x^3+1)))(y-((400)/(x^3)))=0 [0.2, 17.985, -1.124, 7.766]}

(Again multiplied by 100 for clarity.)

And it is not difficult to see that #int_1^oo 4/x^3 dx# converges.

So

#int_1^oo (cos(4x) +3) / (3x^3 + 1) dx# also converges. (To what, I do not know.)