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-

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^{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