# How to test whether an improper integral converges or diverges?

To verify the improper integral $$\int_{1}^{\infty}x\cdot\left | \sin (x^4)\cdot\sin x \right |dx$$ converges or diverges,I code

Integrate[x*Abs[Sin[x^4]*Sin[x]], {x, 1, Infinity}]


then feedback is the expression of $$\int_{1}^{\infty}x\cdot\left | \sin (x^4)\cdot\sin x \right |dx$$.Is that means the improper integral $$\int_{1}^{\infty}x\cdot\left | \sin (x^4)\cdot\sin x \right |dx$$ converges?

• Obviously, the integral diverges. But it is surprising that the system does not give a message, and estimates the numerical value as 1.5408*10^147 – Alex Trounev Jul 3 '19 at 15:33
• @AlexTrounev: Yes,I proved it diverges in the past .But after using mathematica to verify my conclusion, I puzzled ! Thanks for your confirmation ! What's worng with it? – AplehKevin Jul 3 '19 at 15:43
• Not all cases have been studied and not all are entered into the database. – Alex Trounev Jul 3 '19 at 16:27
• @AlexTrounev：Is there any better way to deal with this kind of problem？ – AplehKevin Jul 3 '19 at 16:36
• This is rather a math question than a Mathematica question: see math.stackexchange.com/questions/1018348/… . – user64494 Jul 4 '19 at 17:04