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


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Question Number 84581 by msup trace by abdo last updated on 14/Mar/20

$l e t f (x) = e^{2 x} l n (1 - 3 x^{2})$ $1) c a l c u l a t e f^{(0)} (x) a n d f^{(n)} (0)$ $2) d r v e l o p p f a t i n t e g r s e r i e$ $3) f i n d \int f (x) d x$
Commented by mathmax by abdo last updated on 18/Mar/20
$1) f(x)=e^(2x) ln(1−3x^2 )⇒f^((n)) (x)=Σ_(k=0) ^n C_n ^k (ln(1−3x^2 ))^((k)) (e^(2x) )^((n−k)) =2^n e^(2x) ln(1−3x^2 )+Σ_(k=1) ^n C_n ^k (ln(1−3x^2 ))^((k)) 2^(n−k) e^(2x) we have (ln(1−3x^2 ))^((1)) =((−6x)/(1−3x^2 )) =((6x)/(3x^2 −1)) =((2x)/(x^2 −(1/3))) =(1/(x−(1/(√3))))+(1/(x+(1/(√3)))) ⇒(ln(1−3x^2 ))^((k)) =(((−1)^(k−1) (k−1)!)/((x−(1/(√3)))^k ))+(((−1)^(k−1) (k−1)!)/((x+(1/(√3)))^k )) =(−1)^(k−1) (k−1)!{(((x+(1/(√3)))^k +(x−(1/(√3)))^k )/((x^2 −(1/3))^k ))} ⇒ f^((n)) (x)=2^n e^(2x) ln(1−3x^2 ) +Σ_(k=1) ^n (−1)^(k−1) (k−1)! C_n ^k ×(((x+(1/(√3)))^k +(x−(1/(√3)))^k )/((x^2 −(1/3))^k ))×2^(n−k) e^(2x) f^((n)) (0) =Σ_(k=1) ^n (−1)^(k−1) (k−1)! C_n ^k ((((1/(√3)))^k +(−(1/(√3)))^k )/((−(1/3))^k ))×2^(n−k)$
$1) f (x) = e^{2 x} l n (1 - 3 x^{2}) \Rightarrow f^{(n)} (x) = \sum_{k = 0}^{n} C_{n}^{k} {(l n (1 - 3 x^{2}))}^{(k)} {(e^{2 x})}^{(n - k)}$ $= 2^{n} e^{2 x} l n (1 - 3 x^{2}) + \sum_{k = 1}^{n} C_{n}^{k} {(l n (1 - 3 x^{2}))}^{(k)} 2^{n - k} e^{2 x}$ $w e h a v e {(l n (1 - 3 x^{2}))}^{(1)} = \frac{- 6 x}{1 - 3 x^{2}} = \frac{6 x}{3 x^{2} - 1} = \frac{2 x}{x^{2} - \frac{1}{3}}$ $= \frac{1}{x - \frac{1}{\sqrt{3}}} + \frac{1}{x + \frac{1}{\sqrt{3}}} \Rightarrow {(l n (1 - 3 x^{2}))}^{(k)} = \frac{{(- 1)}^{k - 1} (k - 1)!}{{(x - \frac{1}{\sqrt{3}})}^{k}} + \frac{{(- 1)}^{k - 1} (k - 1)!}{{(x + \frac{1}{\sqrt{3}})}^{k}}$ $= {(- 1)}^{k - 1} (k - 1)! {\frac{{(x + \frac{1}{\sqrt{3}})}^{k} + {(x - \frac{1}{\sqrt{3}})}^{k}}{{(x^{2} - \frac{1}{3})}^{k}}} \Rightarrow$ $f^{(n)} (x) = 2^{n} e^{2 x} l n (1 - 3 x^{2})$ $+ \sum_{k = 1}^{n} {(- 1)}^{k - 1} (k - 1)! C_{n}^{k} \times \frac{{(x + \frac{1}{\sqrt{3}})}^{k} + {(x - \frac{1}{\sqrt{3}})}^{k}}{{(x^{2} - \frac{1}{3})}^{k}} \times 2^{n - k} e^{2 x}$ $f^{(n)} (0) = \sum_{k = 1}^{n} {(- 1)}^{k - 1} (k - 1)! C_{n}^{k} \frac{{(\frac{1}{\sqrt{3}})}^{k} + {(- \frac{1}{\sqrt{3}})}^{k}}{{(- \frac{1}{3})}^{k}} \times 2^{n - k}$
Commented by mathmax by abdo last updated on 18/Mar/20

$2) f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n} a n d f^{(n)} (0) i s k n w n$
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