Question

Determine if the following integral is convergent or divergent.
${\int }_{\infty }^1\frac{1+3si{n}^{4}2x}{\sqrt{x}}$

• A. Convergent
• B. Divergent
• C. Convergent at $1$
  
• D. None of these

Answer 1

The correct option is B Divergent

First notice that as the numerator in this function is going to be bounded since the $sine$ is never larger than $1$. Therefore, since the exponent on the denominator is less than $1$ we can guess that the integral will probably diverge. We will need a smaller function that also diverges

$\frac{1+3si{n}^{4}2x}{\sqrt{x}}<\frac{1}{\sqrt{x}}$

And ${\int }_1^{\infty }\frac{1}{\sqrt{x}}$ always diverges.

so by comparison test we can say that

${\int }_1^{\infty }\frac{1+3si{n}^{4}2x}{\sqrt{x}}dx$ also diverges.