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Question Number 81719 by mathmax by abdo last updated on 14/Feb/20
1) find ∫ (dx/((x+2)^5 (x−3)^9 )) 2) calculate ∫_4 ^(+∞) (dx/((x+2)^5 (x−3)^9 ))
$$\left.\mathrm{1}\right)\:{find}\:\int\:\:\:\frac{{dx}}{\left({x}+\mathrm{2}\right)^{\mathrm{5}} \left({x}−\mathrm{3}\right)^{\mathrm{9}} } \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{4}} ^{+\infty} \:\frac{{dx}}{\left({x}+\mathrm{2}\right)^{\mathrm{5}} \left({x}−\mathrm{3}\right)^{\mathrm{9}} } \\ $$
Commented by mathmax by abdo last updated on 15/Feb/20
let A =∫ (dx/((x+2)^5 (x−3)^9 )) ⇒A =∫ (dx/((((x+2)/(x−3)))^5 (x−3)^(14) )) we use the changement ((x+2)/(x−3)) =t ⇒x+2=tx−3t ⇒(1−t)x=−2−3t ⇒(t−1)x=3t+2 ⇒x=((3t+2)/(t−1)) ⇒x−3=((3t+2)/(t−1))−3 =((3t+2−3t+3)/(t−1)) =(5/(t−1)) ⇒dx =((−5)/((t−1)^2 ))dt ⇒ A = ∫ ((−5)/((t−1)^2 t^5 ((5/(t−1)))^(14) ))dt =((−5)/5^(14) ) ∫ (((t−1)^(14) )/((t−1)^2 t^5 ))dt =−(1/5^(13) ) ∫ (((t−1)^(12) )/t^5 )dt =−(1/5^(13) ) ∫ ((Σ_(k=0) ^(12) C_(12) ^k t^k (−1)^(12−k) )/t^5 )dt =−(1/5^(13) ) ∫ ((C_(12) ^0 −C_(12) ^1 t+C_(12) ^2 t^2 −C_(12) ^3 t^3 +C_(12) ^4 t^4 −C_(12) ^5 t^5 +C_(12) ^6 t^6 −C_(12) ^7 t^7 +C_(12) ^8 t^8 −C_(12) ^9 t^9 +C_(12) ^(10) t^(10) −C_(12) ^(11) t^(11) +C_(12) ^(12) t^(12) )/t^5 )dt =−(1/5^(13) ) ∫((( C_(12) ^0 )/t^5 ) −(C_(12) ^1 /t^4 ) +(C_(12) ^2 /t^3 )−(C_(12) ^3 /t^2 ) +(C_(12) ^4 /t)−C_(12) ^5 +C_(12) ^6 t −C_(12) ^7 t^2 +C_(12) ^8 t^3 −C_(12) ^9 t^4 +C_(12) ^(10) t^5 −C_(12) ^(11) t^6 +C_(12) ^(12) t^7 )dt (−5^(13) )A =−(C_(12) ^0 /(4t^4 ))+(C_(12) ^1 /(3t^3 )) −(C_(12) ^2 /(2t^2 )) +(C_(12) ^3 /t)+C_(12) ^4 ln∣t∣−C_(12) ^5 t+(C_(12) ^6 /2)t^2 −(C_(12) ^7 /3)t^3 +(C_(12) ^8 /4)t^4 −(C_(12) ^9 /5)t^5 +(C_(12) ^(10) /6)t^6 −(C_(12) ^(11) /7)t^7 + (C_(12) ^(12) /8)t^8 +C (−5^(13) )A =−(1/4) C_(12) ^0 (((x−3)/(x+2)))^4 +(1/3)C_(12) ^1 (((x−3)/(x+2)))^3 −(1/2)C_(12) ^2 (((x−3)/(x+2)))^2 +C_(12) ^3 (((x−3)/(x+2))) C_(12) ^4 ln∣((x+2)/(x−3))∣−C_(12) ^5 (((x+2)/(x−3)))+(1/2)C_(12) ^6 (((x+2)/(x−3)))^2 −(1/3)C_(12) ^7 (((x+2)/(x−3)))^3 +(1/4)C_(12) ^8 (((x+2)/(x−3)))^4 −(1/5) C_(12) ^9 (((x+2)/(x−3)))^5 +(1/6) C_(12) ^(10) (((x+2)/(x−3)))^6 −(1/7) C_(12) ^(11) (((x+2)/(x−3)))^7 +(1/8) C_(12) ^(12) (((x+2)/(x−3)))^8 +C so the value of integral A is known
$${let}\:{A}\:=\int\:\:\frac{{dx}}{\left({x}+\mathrm{2}\right)^{\mathrm{5}} \left({x}−\mathrm{3}\right)^{\mathrm{9}} }\:\Rightarrow{A}\:=\int\:\:\frac{{dx}}{\left(\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}\right)^{\mathrm{5}} \left({x}−\mathrm{3}\right)^{\mathrm{14}} } \\ $$$${we}\:{use}\:{the}\:{changement}\:\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}\:={t}\:\Rightarrow{x}+\mathrm{2}={tx}−\mathrm{3}{t}\:\Rightarrow\left(\mathrm{1}−{t}\right){x}=−\mathrm{2}−\mathrm{3}{t} \\ $$$$\Rightarrow\left({t}−\mathrm{1}\right){x}=\mathrm{3}{t}+\mathrm{2}\:\Rightarrow{x}=\frac{\mathrm{3}{t}+\mathrm{2}}{{t}−\mathrm{1}}\:\Rightarrow{x}−\mathrm{3}=\frac{\mathrm{3}{t}+\mathrm{2}}{{t}−\mathrm{1}}−\mathrm{3}\:=\frac{\mathrm{3}{t}+\mathrm{2}−\mathrm{3}{t}+\mathrm{3}}{{t}−\mathrm{1}} \\ $$$$=\frac{\mathrm{5}}{{t}−\mathrm{1}}\:\Rightarrow{dx}\:=\frac{−\mathrm{5}}{\left({t}−\mathrm{1}\right)^{\mathrm{2}} }{dt}\:\Rightarrow \\ $$$${A}\:=\:\int\:\:\frac{−\mathrm{5}}{\left({t}−\mathrm{1}\right)^{\mathrm{2}} \:{t}^{\mathrm{5}} \left(\frac{\mathrm{5}}{{t}−\mathrm{1}}\right)^{\mathrm{14}} }{dt}\:=\frac{−\mathrm{5}}{\mathrm{5}^{\mathrm{14}} }\:\int\:\:\frac{\left({t}−\mathrm{1}\right)^{\mathrm{14}} }{\left({t}−\mathrm{1}\right)^{\mathrm{2}} \:{t}^{\mathrm{5}} }{dt} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{13}} }\:\int\:\:\frac{\left({t}−\mathrm{1}\right)^{\mathrm{12}} }{{t}^{\mathrm{5}} }{dt} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{13}} }\:\int\:\frac{\sum_{{k}=\mathrm{0}} ^{\mathrm{12}} {C}_{\mathrm{12}} ^{{k}} \:{t}^{{k}} \left(−\mathrm{1}\right)^{\mathrm{12}−{k}} }{{t}^{\mathrm{5}} }{dt} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{13}} }\:\int\:\:\frac{{C}_{\mathrm{12}} ^{\mathrm{0}} \:−{C}_{\mathrm{12}} ^{\mathrm{1}} {t}+{C}_{\mathrm{12}} ^{\mathrm{2}} \:{t}^{\mathrm{2}} −{C}_{\mathrm{12}} ^{\mathrm{3}} \:{t}^{\mathrm{3}} +{C}_{\mathrm{12}} ^{\mathrm{4}} \:{t}^{\mathrm{4}} \:−{C}_{\mathrm{12}} ^{\mathrm{5}} \:{t}^{\mathrm{5}} \:+{C}_{\mathrm{12}} ^{\mathrm{6}} \:{t}^{\mathrm{6}} \:−{C}_{\mathrm{12}} ^{\mathrm{7}} \:{t}^{\mathrm{7}} +{C}_{\mathrm{12}} ^{\mathrm{8}} {t}^{\mathrm{8}} \:−{C}_{\mathrm{12}} ^{\mathrm{9}} \:{t}^{\mathrm{9}} \:+{C}_{\mathrm{12}} ^{\mathrm{10}} \:{t}^{\mathrm{10}} \:−{C}_{\mathrm{12}} ^{\mathrm{11}} \:{t}^{\mathrm{11}} +{C}_{\mathrm{12}} ^{\mathrm{12}} \:{t}^{\mathrm{12}} }{{t}^{\mathrm{5}} }{dt}\: \\ $$$$=−\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{13}} }\:\int\left(\frac{\:{C}_{\mathrm{12}} ^{\mathrm{0}} }{{t}^{\mathrm{5}} }\:−\frac{{C}_{\mathrm{12}} ^{\mathrm{1}} }{{t}^{\mathrm{4}} }\:+\frac{{C}_{\mathrm{12}} ^{\mathrm{2}} }{{t}^{\mathrm{3}} }−\frac{{C}_{\mathrm{12}} ^{\mathrm{3}} }{{t}^{\mathrm{2}} }\:+\frac{{C}_{\mathrm{12}} ^{\mathrm{4}} }{{t}}−{C}_{\mathrm{12}} ^{\mathrm{5}} \:+{C}_{\mathrm{12}} ^{\mathrm{6}} \:{t}\:−{C}_{\mathrm{12}} ^{\mathrm{7}} \:{t}^{\mathrm{2}} +{C}_{\mathrm{12}} ^{\mathrm{8}} \:{t}^{\mathrm{3}} −{C}_{\mathrm{12}} ^{\mathrm{9}} \:{t}^{\mathrm{4}} \:+{C}_{\mathrm{12}} ^{\mathrm{10}} \:{t}^{\mathrm{5}} \:−{C}_{\mathrm{12}} ^{\mathrm{11}} \:{t}^{\mathrm{6}} \:+{C}_{\mathrm{12}} ^{\mathrm{12}} \:{t}^{\mathrm{7}} \right){dt} \\ $$$$\left(−\mathrm{5}^{\mathrm{13}} \right){A}\:=−\frac{{C}_{\mathrm{12}} ^{\mathrm{0}} }{\mathrm{4}{t}^{\mathrm{4}} }+\frac{{C}_{\mathrm{12}} ^{\mathrm{1}} }{\mathrm{3}{t}^{\mathrm{3}} }\:−\frac{{C}_{\mathrm{12}} ^{\mathrm{2}} }{\mathrm{2}{t}^{\mathrm{2}} }\:+\frac{{C}_{\mathrm{12}} ^{\mathrm{3}} }{{t}}+{C}_{\mathrm{12}} ^{\mathrm{4}} {ln}\mid{t}\mid−{C}_{\mathrm{12}} ^{\mathrm{5}} \:{t}+\frac{{C}_{\mathrm{12}} ^{\mathrm{6}} }{\mathrm{2}}{t}^{\mathrm{2}} −\frac{{C}_{\mathrm{12}} ^{\mathrm{7}} }{\mathrm{3}}{t}^{\mathrm{3}} \:+\frac{{C}_{\mathrm{12}} ^{\mathrm{8}} }{\mathrm{4}}{t}^{\mathrm{4}} \:−\frac{{C}_{\mathrm{12}} ^{\mathrm{9}} }{\mathrm{5}}{t}^{\mathrm{5}} \:+\frac{{C}_{\mathrm{12}} ^{\mathrm{10}} }{\mathrm{6}}{t}^{\mathrm{6}} \:−\frac{{C}_{\mathrm{12}} ^{\mathrm{11}} }{\mathrm{7}}{t}^{\mathrm{7}} +\:\frac{{C}_{\mathrm{12}} ^{\mathrm{12}} }{\mathrm{8}}{t}^{\mathrm{8}} \:+{C} \\ $$$$\left(−\mathrm{5}^{\mathrm{13}} \right){A}\:=−\frac{\mathrm{1}}{\mathrm{4}}\:{C}_{\mathrm{12}} ^{\mathrm{0}} \:\left(\frac{{x}−\mathrm{3}}{{x}+\mathrm{2}}\right)^{\mathrm{4}} \:+\frac{\mathrm{1}}{\mathrm{3}}{C}_{\mathrm{12}} ^{\mathrm{1}} \left(\frac{{x}−\mathrm{3}}{{x}+\mathrm{2}}\right)^{\mathrm{3}} \:−\frac{\mathrm{1}}{\mathrm{2}}{C}_{\mathrm{12}} ^{\mathrm{2}} \left(\frac{{x}−\mathrm{3}}{{x}+\mathrm{2}}\right)^{\mathrm{2}} \:+{C}_{\mathrm{12}} ^{\mathrm{3}} \left(\frac{{x}−\mathrm{3}}{{x}+\mathrm{2}}\right) \\ $$$${C}_{\mathrm{12}} ^{\mathrm{4}} {ln}\mid\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}\mid−{C}_{\mathrm{12}} ^{\mathrm{5}} \left(\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}\right)+\frac{\mathrm{1}}{\mathrm{2}}{C}_{\mathrm{12}} ^{\mathrm{6}} \left(\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}\right)^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{3}}{C}_{\mathrm{12}} ^{\mathrm{7}} \left(\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}\right)^{\mathrm{3}} \:+\frac{\mathrm{1}}{\mathrm{4}}{C}_{\mathrm{12}} ^{\mathrm{8}} \left(\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}\right)^{\mathrm{4}} \\ $$$$−\frac{\mathrm{1}}{\mathrm{5}}\:{C}_{\mathrm{12}} ^{\mathrm{9}} \:\left(\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}\right)^{\mathrm{5}} +\frac{\mathrm{1}}{\mathrm{6}}\:{C}_{\mathrm{12}} ^{\mathrm{10}} \:\left(\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}\right)^{\mathrm{6}} \:−\frac{\mathrm{1}}{\mathrm{7}}\:{C}_{\mathrm{12}} ^{\mathrm{11}} \:\left(\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}\right)^{\mathrm{7}} \:+\frac{\mathrm{1}}{\mathrm{8}}\:{C}_{\mathrm{12}} ^{\mathrm{12}} \:\left(\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}\right)^{\mathrm{8}} \:+{C} \\ $$$${so}\:{the}\:{value}\:{of}\:{integral}\:{A}\:{is}\:{known} \\ $$$$ \\ $$

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