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Question Number 84581 by msup trace by abdo last updated on 14/Mar/20
let f(x) = e^(2x) ln(1−3x^2 ) 1) calculate f^((0)) (x) and f^((n)) (0) 2) drvelopp f at integr serie 3) find ∫ f(x)dx
$${let}\:{f}\left({x}\right)\:=\:{e}^{\mathrm{2}{x}} {ln}\left(\mathrm{1}−\mathrm{3}{x}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left(\mathrm{0}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{drvelopp}\:{f}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{3}\right)\:{find}\:\int\:{f}\left({x}\right){dx} \\ $$
Commented by mathmax by abdo last updated on 18/Mar/20
1) f(x)=e^(2x) ln(1−3x^2 )⇒f^((n)) (x)=Σ_(k=0) ^n C_n ^k (ln(1−3x^2 ))^((k)) (e^(2x) )^((n−k)) =2^n e^(2x) ln(1−3x^2 )+Σ_(k=1) ^n C_n ^k (ln(1−3x^2 ))^((k)) 2^(n−k) e^(2x) we have (ln(1−3x^2 ))^((1)) =((−6x)/(1−3x^2 )) =((6x)/(3x^2 −1)) =((2x)/(x^2 −(1/3))) =(1/(x−(1/( (√3)))))+(1/(x+(1/( (√3))))) ⇒(ln(1−3x^2 ))^((k)) =(((−1)^(k−1) (k−1)!)/((x−(1/( (√3))))^k ))+(((−1)^(k−1) (k−1)!)/((x+(1/( (√3))))^k )) =(−1)^(k−1) (k−1)!{(((x+(1/( (√3))))^k +(x−(1/( (√3))))^k )/((x^2 −(1/3))^k ))} ⇒ f^((n)) (x)=2^n e^(2x) ln(1−3x^2 ) +Σ_(k=1) ^n (−1)^(k−1) (k−1)! C_n ^k ×(((x+(1/( (√3))))^k +(x−(1/( (√3))))^k )/((x^2 −(1/3))^k ))×2^(n−k) e^(2x) f^((n)) (0) =Σ_(k=1) ^n (−1)^(k−1) (k−1)! C_n ^k ((((1/( (√3))))^k +(−(1/( (√3))))^k )/((−(1/3))^k ))×2^(n−k)
$$\left.\mathrm{1}\right)\:{f}\left({x}\right)={e}^{\mathrm{2}{x}} {ln}\left(\mathrm{1}−\mathrm{3}{x}^{\mathrm{2}} \right)\Rightarrow{f}^{\left({n}\right)} \left({x}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:\left({ln}\left(\mathrm{1}−\mathrm{3}{x}^{\mathrm{2}} \right)\right)^{\left({k}\right)} \left({e}^{\mathrm{2}{x}} \right)^{\left({n}−{k}\right)} \\ $$$$=\mathrm{2}^{{n}} \:{e}^{\mathrm{2}{x}} {ln}\left(\mathrm{1}−\mathrm{3}{x}^{\mathrm{2}} \right)+\sum_{{k}=\mathrm{1}} ^{{n}} \:{C}_{{n}} ^{{k}} \:\:\left({ln}\left(\mathrm{1}−\mathrm{3}{x}^{\mathrm{2}} \right)\right)^{\left({k}\right)} \:\mathrm{2}^{{n}−{k}} \:{e}^{\mathrm{2}{x}} \\ $$$${we}\:{have}\:\left({ln}\left(\mathrm{1}−\mathrm{3}{x}^{\mathrm{2}} \right)\right)^{\left(\mathrm{1}\right)} =\frac{−\mathrm{6}{x}}{\mathrm{1}−\mathrm{3}{x}^{\mathrm{2}} }\:=\frac{\mathrm{6}{x}}{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{1}}\:=\frac{\mathrm{2}{x}}{{x}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{3}}} \\ $$$$=\frac{\mathrm{1}}{{x}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}}+\frac{\mathrm{1}}{{x}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}}\:\Rightarrow\left({ln}\left(\mathrm{1}−\mathrm{3}{x}^{\mathrm{2}} \right)\right)^{\left({k}\right)} =\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!}{\left({x}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)^{{k}} }+\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!}{\left({x}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)^{{k}} } \\ $$$$=\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!\left\{\frac{\left({x}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)^{{k}} +\left({x}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)^{{k}} }{\left({x}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{3}}\right)^{{k}} }\right\}\:\Rightarrow \\ $$$${f}^{\left({n}\right)} \left({x}\right)=\mathrm{2}^{{n}} \:{e}^{\mathrm{2}{x}} {ln}\left(\mathrm{1}−\mathrm{3}{x}^{\mathrm{2}} \right) \\ $$$$+\sum_{{k}=\mathrm{1}} ^{{n}} \:\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!\:{C}_{{n}} ^{{k}} \:\:\:×\frac{\left({x}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)^{{k}} \:+\left({x}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)^{{k}} }{\left({x}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{3}}\right)^{{k}} }×\mathrm{2}^{{n}−{k}} \:{e}^{\mathrm{2}{x}} \\ $$$${f}^{\left({n}\right)} \left(\mathrm{0}\right)\:=\sum_{{k}=\mathrm{1}} ^{{n}} \left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!\:{C}_{{n}} ^{{k}} \:\frac{\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)^{{k}} \:+\left(−\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)^{{k}} }{\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)^{{k}} }×\mathrm{2}^{{n}−{k}} \\ $$
Commented by mathmax by abdo last updated on 18/Mar/20
2) f(x) =Σ_(n=0) ^∞ ((f^((n)) (0))/(n!)) x^n and f^((n)) (0) is knwn
$$\left.\mathrm{2}\right)\:{f}\left({x}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{f}^{\left({n}\right)} \left(\mathrm{0}\right)}{{n}!}\:{x}^{{n}} \:\:\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right)\:{is}\:{knwn} \\ $$

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