let f(x) =arctan(x^3 ) 1)calculate f^((n)) (x)and f^((n)) (0) 2) developp f at integr serie 3) calculate ∫


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Question Number 67187 by mathmax by abdo last updated on 23/Aug/19

$l e t f (x) = a r c t a n (x^{3})$ $1) c a l c u l a t e f^{(n)} (x) a n d f^{(n)} (0)$ $2) d e v e l o p p f a t i n t e g r s e r i e$ $3) c a l c u l a t e \int_{0}^{1} a r c t a n (x^{3}) d x$
Commented by mathmax by abdo last updated on 28/Aug/19
$1) f(x) =arctan(x^3 ) ⇒f^′ (x) =((3x^2 )/(1+x^6 )) ⇒f^((n)) (x)=(((3x^2 )/(1+x^6 )))^((n−1)) let decompose inside C(x) F(x) =((3x^2 )/(x^6 +1)) z^6 +1 =0 ⇒z^6 =−1 =e^(i(2k+1)π) so if z=r e^(iθ) we get r=1 and 6θ =e^(i(2k+1)π) ⇒ θ =e^((i(2k+1)π)/6) so the poles of F are Z_k =e^((i(2k+1)π)/6) k∈[[0,5]] Z_0 =e^((iπ)/6) , Z_1 =e^(i(π/2)) =i Z_2 =e^((i5π)/6) , Z_3 = e^((i7π)/6) , Z_4 = e^((i9π)/6) , Z_5 = e^((i11π)/6) f^((n)) (x) =Σ_(k=0) ^(n−1) C_(n−1) ^k (3x^2 )^((k)) ((1/(x^6 +1)))^((n−1−k)) (leibniz) =3x^2 ×{(1/(x^6 +1))}^((n−1)) +(n−1)(6x){(1/(x^6 +1))}^((n−2)) +6 C_(n−1) ^2 {(1/(x^6 +1))}^((n−3)) let find {(1/(x^6 +1))}^((k)) with k from N wr have G(x) =(1/(x^6 +1)) =Σ_(i=0) ^5 (λ_i /(x−z_i )) λ_i =(1/(6z_i ^5 )) =−(1/6)z_i ⇒G(x) =−(1/6)Σ_(i=0) ^5 (z_i /(x−z_i )) ⇒ G^((k)) (x) =−(1/6)Σ_(i=0) ^5 z_i ×(((−1)^k k!)/((x−z_i )^(k+1) )) ⇒ G^((n−1)) (x) =−(1/6)Σ_(i=0) ^5 z_i ×(((−1)^(n−1) (n−1)!)/((x−z_i )^n )) G^((n−2)) (x) =−(1/6)Σ_(i=0) ^5 z_i × (((−1)^(n−2) (n−2)!)/((x−z_i )^(n−1) )) G^((n−3)) (x) =−(1/6)Σ_(i=0) ^5 z_i ×(((−1)^(n−3) (n−3)!)/((x−z_i )^(n−2) )) so the value of f^((n)) (x) is determined.$
$1) f (x) = a r c t a n (x^{3}) \Rightarrow f^{^{'}} (x) = \frac{3 x^{2}}{1 + x^{6}} \Rightarrow f^{(n)} (x) = {(\frac{3 x^{2}}{1 + x^{6}})}^{(n - 1)}$ $l e t d e c o m p o s e i n s i d e C (x) F (x) = \frac{3 x^{2}}{x^{6} + 1}$ $z^{6} + 1 = 0 \Rightarrow z^{6} = - 1 = e^{i (2 k + 1) π} s o i f z = r e^{i θ} w e g e t r = 1 a n d$ $6 θ = e^{i (2 k + 1) π} \Rightarrow θ = e^{\frac{i (2 k + 1) π}{6}} s o t h e p o l e s o f F a r e$ $Z_{k} = e^{\frac{i (2 k + 1) π}{6}} k \in [[0, 5]] Z_{0} = e^{\frac{i π}{6}}, Z_{1} = e^{i \frac{π}{2}} = i$ $Z_{2} = e^{\frac{i 5 π}{6}}, Z_{3} = e^{\frac{i 7 π}{6}}, Z_{4} = e^{\frac{i 9 π}{6}}, Z_{5} = e^{\frac{i 11 π}{6}}$ $f^{(n)} (x) = \sum_{k = 0}^{n - 1} C_{n - 1}^{k} {(3 x^{2})}^{(k)} {(\frac{1}{x^{6} + 1})}^{(n - 1 - k)} (l e i b n i z)$ $= 3 x^{2} \times {\frac{1}{x^{6} + 1}}^{(n - 1)} + (n - 1) (6 x) {\frac{1}{x^{6} + 1}}^{(n - 2)}$ $+ 6 C_{n - 1}^{2} {\frac{1}{x^{6} + 1}}^{(n - 3)} l e t f i n d {\frac{1}{x^{6} + 1}}^{(k)} w i t h k f r o m N$ $w r h a v e G (x) = \frac{1}{x^{6} + 1} = \sum_{i = 0}^{5} \frac{λ_{i}}{x - z_{i}}$ $λ_{i} = \frac{1}{6 z_{i}^{5}} = - \frac{1}{6} z_{i} \Rightarrow G (x) = - \frac{1}{6} \sum_{i = 0}^{5} \frac{z_{i}}{x - z_{i}} \Rightarrow$ $G^{(k)} (x) = - \frac{1}{6} \sum_{i = 0}^{5} z_{i} \times \frac{{(- 1)}^{k} k!}{{(x - z_{i})}^{k + 1}} \Rightarrow$ $G^{(n - 1)} (x) = - \frac{1}{6} \sum_{i = 0}^{5} z_{i} \times \frac{{(- 1)}^{n - 1} (n - 1)!}{{(x - z_{i})}^{n}}$ $G^{(n - 2)} (x) = - \frac{1}{6} \sum_{i = 0}^{5} z_{i} \times \frac{{(- 1)}^{n - 2} (n - 2)!}{{(x - z_{i})}^{n - 1}}$ $G^{(n - 3)} (x) = - \frac{1}{6} \sum_{i = 0}^{5} z_{i} \times \frac{{(- 1)}^{n - 3} (n - 3)!}{{(x - z_{i})}^{n - 2}} s o t h e v a l u e o f$ $f^{(n)} (x) i s d e t e r m i n e d .$
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