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Question Number 140823 by liberty last updated on 13/May/21

$$\mathrm{Determine}\mathrm{whether}\mathrm{the}\mathrm{improper}$$$$\mathrm{integral}\mathrm{converges}\mathrm{or}\mathrm{diverges}$$$${\int }_1^{\mathrm{\infty }}\frac{2\mathrm{x}+7}{{7\mathrm{x}}^3+{5\mathrm{x}}^2+1}\mathrm{dx}$$

Commented by liberty last updated on 13/May/21

$$$$$$\mathrm{can}\mathrm{anyone}\mathrm{help}\mathrm{me}$$

Answered by mathmax by abdo last updated on 13/May/21

$$\mathrm{x}\to \frac{2\mathrm{x}+7}{{7\mathrm{x}}^3+{5\mathrm{x}}^2+1}\mathrm{continue}\mathrm{on}[1,\mathrm{a}[\mathrm{so}\mathrm{integrable}\mathrm{on}[1,\mathrm{a}[(\mathrm{a}>1)$$$$\mathrm{at}+\mathrm{\infty }\frac{2\mathrm{x}+7}{{7\mathrm{x}}^3+{5\mathrm{x}}^2+1}\sim \frac{2}{{7\mathrm{x}}^2}\mathrm{and}{\int }_{\mathrm{a}}^{\mathrm{\infty }}\frac{2}{{7\mathrm{x}}^2}\mathrm{dx}\mathrm{cv}\Rightarrow \mathrm{this}\mathrm{integral}\mathrm{is}\mathrm{cv}.$$

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