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Question Number 68238 by mathmax by abdo last updated on 07/Sep/19
let f(x)=(x^2 −3x)arctan(2x+1) 1) determine f^((n)) (x) and f^((n)) (0) 2)developp f at integr serie 3) calculate ∫_0 ^1 f(x)dx
$${let}\:\:{f}\left({x}\right)=\left({x}^{\mathrm{2}} −\mathrm{3}{x}\right){arctan}\left(\mathrm{2}{x}+\mathrm{1}\right) \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{f}^{\left({n}\right)} \left({x}\right)\:\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$
Commented by mathmax by abdo last updated on 08/Sep/19
2) f(x) =Σ_(n=0) ^∞ ((f^((n)) (0))/(n!)) x^n =f(0) +Σ_(n=1) ^∞ ((f^((n)) (0))/(n!)) x^n 1)f(x)=(x^2 −3x)arctan(2x+1) leibniz formula give f^((n)) (x) =Σ_(k=0) ^n C_n ^k (x^2 −3x)^((k)) (arctan(2x+1))^((n−k)) =(x^2 −3x)(arctan(2x+1))^((n)) +n(2x−3)(arctan(2x+1))^((n−1)) +n(n−1)(arctan(2x+1))^((n−2)) let A(x)=arctan(2x+1) ⇒ A^′ (x) =(2/(1+(2x+1)^2 )) =(2/(1+4x^2 +4x +1)) =(2/(4x^2 +4x +2)) =(1/(2x^2 +2x+1)) ⇒A^((n)) (x)=((1/(2x^2 +2x+1)))^((n−1)) 2x^2 +2x+1 =0→Δ^′ =1−2 =−1 ⇒x_1 =((−1+i)/2) =−((1−i)/2)=−((√2)/2)e^(−((iπ)/4)) x_2 =((−1−i)/2) =−((√2)/2)e^((iπ)/4) ⇒ (1/(2x^2 +2x+1)) =(1/(2(x+((√2)/2)e^(−((iπ)/4)) )(x+((√2)/2)e^((iπ)/4) ))) =(1/(2i)){ (1/(x+((√2)/2)e^(−((iπ)/4)) ))−(1/(x+((√2)/2)e^((iπ)/4) ))} ⇒ A^((n)) =(1/(2i)){ (((−1)^(n−1) (n−1)!)/((x+((√2)/2)e^(−((iπ)/4)) )^n ))−(((−1)^(n−1) (n−1)!)/((x+((√2)/2)e^((iπ)/4) )^n ))} =(((−1)^(n−1) (n−1)!)/(2i)){(((x+((√2)/2)e^((iπ)/4) )^n −(x+((√2)/2)e^(−((iπ)/4)) )^n )/((x^2 +x+(1/2))^n ))} =(((−1)^(n−1) (n−1)!)/(2i))×((2i Im((x+((√2)/2)e^((iπ)/4) )^n ))/((x^2 +x+(1/2))^n )) ⇒ A^((n)) (x)=(((−1)^(n−1) (n−1)!Im((x+((√2)/2)e^((iπ)/4) )^n ))/((x^2 +x+(1/2))^n )) ⇒ f^((n)) (x) =(x^2 −3x)A^((n)) (x)+n(2x−3) A^((n−1)) (x)+n(n−1)A^((n−2)) (x) ⇒f^((n)) (0) =−3n A^((n−1)) (0)+n(n−1) A^((n−2)) (0) A^((n)) (0) =2^n (−1)^(n−1) (n−1)!Im(((√2)/2)e^((iπ)/4) )^n =2^n (−1)^(n−1) (n−1)!(2^(n/2) /2^n )sin(n(π/4)) =2^(n/2) (−1)^(n−1) (n−1)!sin(((nπ)/4)) ⇒ f^((n)) (0) =−3n 2^((n−1)/2) (−1)^(n−2) (n−2)!sin((((n−1)π)/4)) +n(n−1)2^((n−2)/2) (−1)^(n−3) (n−3)!sin((((n−2)π)/4))
$$\left.\mathrm{2}\right)\:{f}\left({x}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{f}^{\left({n}\right)} \left(\mathrm{0}\right)}{{n}!}\:{x}^{{n}} \:={f}\left(\mathrm{0}\right)\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{f}^{\left({n}\right)} \left(\mathrm{0}\right)}{{n}!}\:{x}^{{n}} \\ $$$$\left.\mathrm{1}\right){f}\left({x}\right)=\left({x}^{\mathrm{2}} −\mathrm{3}{x}\right){arctan}\left(\mathrm{2}{x}+\mathrm{1}\right)\:\:{leibniz}\:{formula}\:{give} \\ $$$${f}^{\left({n}\right)} \left({x}\right)\:=\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \left({x}^{\mathrm{2}} −\mathrm{3}{x}\right)^{\left({k}\right)} \left({arctan}\left(\mathrm{2}{x}+\mathrm{1}\right)\right)^{\left({n}−{k}\right)} \\ $$$$=\left({x}^{\mathrm{2}} −\mathrm{3}{x}\right)\left({arctan}\left(\mathrm{2}{x}+\mathrm{1}\right)\right)^{\left({n}\right)} \:+{n}\left(\mathrm{2}{x}−\mathrm{3}\right)\left({arctan}\left(\mathrm{2}{x}+\mathrm{1}\right)\right)^{\left({n}−\mathrm{1}\right)} \\ $$$$+{n}\left({n}−\mathrm{1}\right)\left({arctan}\left(\mathrm{2}{x}+\mathrm{1}\right)\right)^{\left({n}−\mathrm{2}\right)} \:{let}\:{A}\left({x}\right)={arctan}\left(\mathrm{2}{x}+\mathrm{1}\right)\:\Rightarrow \\ $$$${A}^{'} \left({x}\right)\:=\frac{\mathrm{2}}{\mathrm{1}+\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{2}} }\:=\frac{\mathrm{2}}{\mathrm{1}+\mathrm{4}{x}^{\mathrm{2}} \:+\mathrm{4}{x}\:+\mathrm{1}}\:=\frac{\mathrm{2}}{\mathrm{4}{x}^{\mathrm{2}} \:+\mathrm{4}{x}\:+\mathrm{2}}\:=\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} \:+\mathrm{2}{x}+\mathrm{1}} \\ $$$$\Rightarrow{A}^{\left({n}\right)} \left({x}\right)=\left(\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} \:+\mathrm{2}{x}+\mathrm{1}}\right)^{\left({n}−\mathrm{1}\right)} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} \:+\mathrm{2}{x}+\mathrm{1}\:=\mathrm{0}\rightarrow\Delta^{'} =\mathrm{1}−\mathrm{2}\:=−\mathrm{1}\:\Rightarrow{x}_{\mathrm{1}} =\frac{−\mathrm{1}+{i}}{\mathrm{2}}\:=−\frac{\mathrm{1}−{i}}{\mathrm{2}}=−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{e}^{−\frac{{i}\pi}{\mathrm{4}}} \\ $$$${x}_{\mathrm{2}} =\frac{−\mathrm{1}−{i}}{\mathrm{2}}\:=−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{4}}} \:\Rightarrow \\ $$$$\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} \:+\mathrm{2}{x}+\mathrm{1}}\:=\frac{\mathrm{1}}{\mathrm{2}\left({x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{e}^{−\frac{{i}\pi}{\mathrm{4}}} \right)\left({x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{4}}} \right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}{i}}\left\{\:\:\frac{\mathrm{1}}{{x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{e}^{−\frac{{i}\pi}{\mathrm{4}}} }−\frac{\mathrm{1}}{{x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{4}}} }\right\}\:\Rightarrow \\ $$$${A}^{\left({n}\right)} =\frac{\mathrm{1}}{\mathrm{2}{i}}\left\{\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!}{\left({x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{e}^{−\frac{{i}\pi}{\mathrm{4}}} \right)^{{n}} }−\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!}{\left({x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{4}}} \right)^{{n}} }\right\} \\ $$$$=\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!}{\mathrm{2}{i}}\left\{\frac{\left({x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{4}}} \right)^{{n}} −\left({x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{e}^{−\frac{{i}\pi}{\mathrm{4}}} \right)^{{n}} }{\left({x}^{\mathrm{2}} \:+{x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{{n}} }\right\} \\ $$$$=\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!}{\mathrm{2}{i}}×\frac{\mathrm{2}{i}\:{Im}\left(\left({x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{4}}} \right)^{{n}} \right)}{\left({x}^{\mathrm{2}} \:+{x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{{n}} }\:\Rightarrow \\ $$$${A}^{\left({n}\right)} \left({x}\right)=\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!{Im}\left(\left({x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{4}}} \right)^{{n}} \right)}{\left({x}^{\mathrm{2}} \:+{x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{{n}} }\:\Rightarrow \\ $$$${f}^{\left({n}\right)} \left({x}\right)\:=\left({x}^{\mathrm{2}} −\mathrm{3}{x}\right){A}^{\left({n}\right)} \left({x}\right)+{n}\left(\mathrm{2}{x}−\mathrm{3}\right)\:{A}^{\left({n}−\mathrm{1}\right)} \left({x}\right)+{n}\left({n}−\mathrm{1}\right){A}^{\left({n}−\mathrm{2}\right)} \left({x}\right) \\ $$$$\Rightarrow{f}^{\left({n}\right)} \left(\mathrm{0}\right)\:=−\mathrm{3}{n}\:{A}^{\left({n}−\mathrm{1}\right)} \left(\mathrm{0}\right)+{n}\left({n}−\mathrm{1}\right)\:{A}^{\left({n}−\mathrm{2}\right)} \left(\mathrm{0}\right) \\ $$$${A}^{\left({n}\right)} \left(\mathrm{0}\right)\:=\mathrm{2}^{{n}} \left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!{Im}\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{4}}} \right)^{{n}} \\ $$$$=\mathrm{2}^{{n}} \left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!\frac{\mathrm{2}^{\frac{{n}}{\mathrm{2}}} }{\mathrm{2}^{{n}} }{sin}\left({n}\frac{\pi}{\mathrm{4}}\right)\:=\mathrm{2}^{\frac{{n}}{\mathrm{2}}} \left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!{sin}\left(\frac{{n}\pi}{\mathrm{4}}\right)\:\Rightarrow \\ $$$${f}^{\left({n}\right)} \left(\mathrm{0}\right)\:=−\mathrm{3}{n}\:\mathrm{2}^{\frac{{n}−\mathrm{1}}{\mathrm{2}}} \left(−\mathrm{1}\right)^{{n}−\mathrm{2}} \left({n}−\mathrm{2}\right)!{sin}\left(\frac{\left({n}−\mathrm{1}\right)\pi}{\mathrm{4}}\right) \\ $$$$+{n}\left({n}−\mathrm{1}\right)\mathrm{2}^{\frac{{n}−\mathrm{2}}{\mathrm{2}}} \left(−\mathrm{1}\right)^{{n}−\mathrm{3}} \left({n}−\mathrm{3}\right)!{sin}\left(\frac{\left({n}−\mathrm{2}\right)\pi}{\mathrm{4}}\right) \\ $$$$ \\ $$
Commented by mathmax by abdo last updated on 08/Sep/19
2) f(x) =Σ_(n=0) ^∞ ((f^((n)) (0))/(n!)) x^n =f(0)+((f^((1)) (0))/(1!)) x +((f^((2)) (0))/(2!)) x^2 +Σ_(n=3) ^∞ ((f^((n)) (0))/(n!)) x^n =f^′ (0)x +((f^((2)) (0))/2)x^2 +Σ_(n=3) ^∞ (((−3n2^((n−1)/2) (−1)^(n−2) (n−2)!sin((((n−1)π)/4))+n(n−1)2^((n−2)/2) (−1)^(n−3) (n−3)!sin((((n−2)π)/4)))/(n!)) )x^n
$$\left.\mathrm{2}\right)\:{f}\left({x}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{f}^{\left({n}\right)} \left(\mathrm{0}\right)}{{n}!}\:{x}^{{n}} \\ $$$$={f}\left(\mathrm{0}\right)+\frac{{f}^{\left(\mathrm{1}\right)} \left(\mathrm{0}\right)}{\mathrm{1}!}\:{x}\:+\frac{{f}^{\left(\mathrm{2}\right)} \left(\mathrm{0}\right)}{\mathrm{2}!}\:{x}^{\mathrm{2}} \:+\sum_{{n}=\mathrm{3}} ^{\infty} \:\:\frac{{f}^{\left({n}\right)} \left(\mathrm{0}\right)}{{n}!}\:{x}^{{n}} \\ $$$$={f}^{'} \left(\mathrm{0}\right){x}\:+\frac{{f}^{\left(\mathrm{2}\right)} \left(\mathrm{0}\right)}{\mathrm{2}}{x}^{\mathrm{2}} \: \\ $$$$+\sum_{{n}=\mathrm{3}} ^{\infty} \:\:\left(\frac{−\mathrm{3}{n}\mathrm{2}^{\frac{{n}−\mathrm{1}}{\mathrm{2}}} \left(−\mathrm{1}\right)^{\boldsymbol{{n}}−\mathrm{2}} \left(\boldsymbol{{n}}−\mathrm{2}\right)!{sin}\left(\frac{\left({n}−\mathrm{1}\right)\pi}{\mathrm{4}}\right)+{n}\left({n}−\mathrm{1}\right)\mathrm{2}^{\frac{{n}−\mathrm{2}}{\mathrm{2}}} \left(−\mathrm{1}\right)^{\boldsymbol{{n}}−\mathrm{3}} \left(\boldsymbol{{n}}−\mathrm{3}\right)!{sin}\left(\frac{\left({n}−\mathrm{2}\right)\pi}{\mathrm{4}}\right)}{{n}!}\:\right){x}^{{n}} \\ $$

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