$Is $\int_1^{\infty}\frac{x+\sqrt{x}+\sin x}{x^2-x+1}dx$ convergent?$

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$Is $\int_1^{\infty}\frac{x+\sqrt{x}+\sin x}{x^2-x+1}dx$ convergent?$

Determine whether $\int_1^{\infty}\frac{x+\sqrt{x}+\sin x}{x^2-x+1}dx$ is convergent or divergent. Fully justify your answer.

Calculus Integrals

Lilian00

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Note that
\[\lim_{x \rightarrow \infty} \frac{\frac{x+\sqrt{x}+\sin x}{x^2-x+1}}{\frac{1}{x}}=\lim_{x \rightarrow \infty} \frac{\frac{x}{x^2}.\frac{1+\frac{\sqrt{x}}{x}+\frac{\sin x}{x}}{1-1/x+1/x^2}}{\frac{1}{x}}=1.\]

Thus by the limit comparision theorem, the convergence behavior of this integral is the same as
\[\int_1^{\infty}\frac{1}{x}dx\]
which is divergent. So the given improper integral is divergent.

Nicolasp

163

$Is $\int_1^{\infty}\frac{x+\sqrt{x}+\sin x}{x^2-x+1}dx$ convergent?$

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