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Question Number 36541 by tanmay.chaudhury50@gmail.com last updated on 03/Jun/18

∫(x+(√(1+x^2 )) )^n dx

$$\int\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:\right)^{{n}} {dx} \\ $$

Commented by prof Abdo imad last updated on 03/Jun/18

let put A_n = ∫ (x +(√(1+x^2 )) )^n dx changement x =sh(t) give A_n = ∫ (sht +cht)^n ch dt =∫ { ((e^t −e^(−t) + e^t +e^(−t) )/2)}^n cht dt =∫ e^(nt) ( ((e^t +e^(−t) )/2))dt =(1/2)∫ e^((n+1)t) dt +(1/2)∫ e^((n−1)t) dt = (1/(2(n+1)))e^((n+1)t) +(1/(2(n+1)))e^((n−1)t) +c but e^((n+1)t) =e^((n+1)ln(x+(√(1+x^2 )))) =(x +(√(1+x^2 )))^(n+1) and by?tha same e^((n−1)t) = (x+(√(1+x^2 )))^(n−1) ⇒ A_n = (1/(2(n+1))){ x+(√(1+x^2 )))^(n+1) + (1/(2(n−1))){ x +(√(1+x^2 )))^(n−1) +c if n≠1

$${let}\:{put}\:\:{A}_{{n}} =\:\int\:\left({x}\:+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:\right)^{{n}} \:{dx}\:{changement} \\ $$$${x}\:={sh}\left({t}\right)\:{give}\: \\ $$$${A}_{{n}} =\:\int\:\left({sht}\:\:+{cht}\right)^{{n}} \:{ch}\:{dt} \\ $$$$=\int\:\left\{\:\frac{{e}^{{t}} \:−{e}^{−{t}} \:+\:{e}^{{t}} \:+{e}^{−{t}} }{\mathrm{2}}\right\}^{{n}} \:{cht}\:{dt} \\ $$$$=\int\:\:{e}^{{nt}} \:\left(\:\frac{{e}^{{t}} \:+{e}^{−{t}} }{\mathrm{2}}\right){dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\:\:{e}^{\left({n}+\mathrm{1}\right){t}} {dt}\:\:\:+\frac{\mathrm{1}}{\mathrm{2}}\int\:\:{e}^{\left({n}−\mathrm{1}\right){t}} {dt} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)}{e}^{\left({n}+\mathrm{1}\right){t}} \:\:\:+\frac{\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)}{e}^{\left({n}−\mathrm{1}\right){t}} \:\:+{c} \\ $$$${but}\:{e}^{\left({n}+\mathrm{1}\right){t}} \:={e}^{\left({n}+\mathrm{1}\right){ln}\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)} \\ $$$$=\left({x}\:+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)^{{n}+\mathrm{1}} \:{and}\:{by}?{tha}\:{same} \\ $$$${e}^{\left({n}−\mathrm{1}\right){t}} \:=\:\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)^{{n}−\mathrm{1}} \:\:\Rightarrow \\ $$$${A}_{{n}} \:\:=\:\frac{\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)}\left\{\:{x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)^{{n}+\mathrm{1}} \\ $$$$+\:\frac{\mathrm{1}}{\mathrm{2}\left({n}−\mathrm{1}\right)}\left\{\:{x}\:+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)^{{n}−\mathrm{1}} \:\:+{c}\:\:\:\:{if}\:{n}\neq\mathrm{1} \\ $$

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