Question

choose your favourate colour among the three
and show that it is convergent

Answer 1

\[\huge \color{green} {\int _0^{\infty} \frac{dx}{1+x^3sin^2 x}}\]

Answer 2

\[\huge \color{red} {\int\limits _0^\infty \left\{ \frac{\ln(1+x)}{x}\right\}^2}\]

Answer 3

\[\huge \color{blue} {\int\limits _0^{\infty} \sin (x^p) dx \text{ where p>1}} \]

Answer 4

very smart statement!!!

Answer 5

yw thank you

Answer 6

@RadEn

Answer 7

for blue ı know \[x^p\]
dıverges

Answer 8

indefinite integrals. awesome.
Simply evaluate the integral from 0<x<B
where "B" is some constant. Then, take the limit as B-> 0
if the value exists, then it converges, if otherwise, it's divergent.

for instance, if your answer fot this was\(2\sqrt{5}\) it would be convergent. and a value of \(\pm \infty\) would be divergent.

also FACT: \[\int\limits_{0}^{B}\frac{ 1 }{ x^C }dx \]
it is said to be convergent if C > 1 and divergent if otherwise (meaning p<1)