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Question Number 140823 by liberty last updated on 13/May/21
Determine whether the improper  integral converges or diverges   ∫_1 ^( ∞)  ((2x+7)/(7x^3 +5x^2 +1)) dx
$$\mathrm{Determine}\:\mathrm{whether}\:\mathrm{the}\:\mathrm{improper} \\ $$$$\mathrm{integral}\:\mathrm{converges}\:\mathrm{or}\:\mathrm{diverges}\: \\ $$$$\int_{\mathrm{1}} ^{\:\infty} \:\frac{\mathrm{2x}+\mathrm{7}}{\mathrm{7x}^{\mathrm{3}} +\mathrm{5x}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{dx}\: \\ $$
Commented by liberty last updated on 13/May/21
       can anyone help me
$$ \\ $$$$\:\:\:\:\:\mathrm{can}\:\mathrm{anyone}\:\mathrm{help}\:\mathrm{me} \\ $$
Answered by mathmax by abdo last updated on 13/May/21
x→((2x+7)/(7x^3  +5x^2  +1)) continue on [1,a[  so integrable on [1,a[  (a>1)  at +∞   ((2x+7)/(7x^3  +5x^2  +1))∼(2/(7x^2 )) and ∫_a ^∞  (2/(7x^2 ))dx cv ⇒this integral is cv.
$$\mathrm{x}\rightarrow\frac{\mathrm{2x}+\mathrm{7}}{\mathrm{7x}^{\mathrm{3}} \:+\mathrm{5x}^{\mathrm{2}} \:+\mathrm{1}}\:\mathrm{continue}\:\mathrm{on}\:\left[\mathrm{1},\mathrm{a}\left[\:\:\mathrm{so}\:\mathrm{integrable}\:\mathrm{on}\:\left[\mathrm{1},\mathrm{a}\left[\:\:\left(\mathrm{a}>\mathrm{1}\right)\right.\right.\right.\right. \\ $$$$\mathrm{at}\:+\infty\:\:\:\frac{\mathrm{2x}+\mathrm{7}}{\mathrm{7x}^{\mathrm{3}} \:+\mathrm{5x}^{\mathrm{2}} \:+\mathrm{1}}\sim\frac{\mathrm{2}}{\mathrm{7x}^{\mathrm{2}} }\:\mathrm{and}\:\int_{\mathrm{a}} ^{\infty} \:\frac{\mathrm{2}}{\mathrm{7x}^{\mathrm{2}} }\mathrm{dx}\:\mathrm{cv}\:\Rightarrow\mathrm{this}\:\mathrm{integral}\:\mathrm{is}\:\mathrm{cv}. \\ $$

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