How to find out whether that integral is convergent or divergent?

Question

integral from 0 to infinity ( (sinx+cosx)/(1+x^3) ) dx

 

there's no need in finding improper integral itself, just whether it's convergent or no.

Answer 1

The integral is convergent by the comparison test.

Since |sinx + cosx| ≤ 2

∫₀^∞(sinx+cosx)/(1+x³) dx

≤ 2∫₀^∞ 1/(1+x³) dx

= 2∫₀¹ 1/(1+x³) dx + 2∫₁^∞ 1/(1+x³) dx. 

< 2∫₀¹ 1/(1+x³) dx + 2∫₁^∞ 1/x³ dx. 

 

Since ∫₀¹ 1/(1+x³ )dx is finite, and ∫₁^∞ 1/x³ dx converges, 

{Remember that ∫₁^∞ 1/x^p dx converges to 1/(p-1) for p>1?}

 

∫₀^∞(sinx+cosx)/(1+x³) dx converges

Enjoy!