Question

There are four integrals:
1) definite integral x/(1+x^4)dx b/w 0_infinity

2) definite integral (x^2)/(1+x^4)dx b/w 0_infinity

3) definite integral (x^3)/(1+x^4)dx b/w 0_infinity

4) definite integral (x^4)/(1+x^4)dx b/w 0_infinity

Which of these integrals converge. First of all, what does it mean "converge"? How do you compare to "pure" powers of x?

How would you compute the exact value of at least one of the convergent integrals?

thanks

Answer 1

examine the powers.

4) x^4 dx is an x^5 in the numerator, an x^4 in the denominator. It cant converge to a finite value at inf.
1,2) definitely do, they converge to zero at inf
3) Think on that.

Answer 2

do you do u substitution for each or automatically trig substitution. Because when you do trig sub you get 0 but when you do u sub then you get pi/4