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Question Number 86671 by M±th+et£s last updated on 30/Mar/20

∫sin(x) arcsin(x)

$$\int{sin}\left({x}\right)\:{arcsin}\left({x}\right) \\ $$

Answered by Rio Michael last updated on 30/Mar/20

let me give a try. ∫ sin (x) arcsin(x) dx using the taylor series expansion for sin (x) arcsin (x) centred at 0 sin (x) arcsin (x) = x^2 + (x^6 /(18)) + (x^8 /(30)) + ((2669 x^(10) )/(113400)) + ((601 x^(12) )/(34020)) + ... ⇒ ∫ sin (x) arcsin (x) dx = ∫(x^2 + (x^6 /(18)) + (x^8 /(30)) + ((2669 x^(10) )/(113400)) + ((601 x^(12) )/(34020))+...)dx = (x^3 /3) + (x^7 /(126)) + (x^9 /(270)) + ((2669 x^(11) )/(1247400)) + ((601 x^(13) )/(442260)) + ... + k i still personally think this problem will be cool if we had boundary conditions like ∫_0 ^1 sin (x) arcsin (x) dx.

$$\mathrm{let}\:\mathrm{me}\:\mathrm{give}\:\mathrm{a}\:\mathrm{try}. \\ $$$$\:\int\:\mathrm{sin}\:\left({x}\right)\:\mathrm{arcsin}\left({x}\right)\:{dx} \\ $$$$\mathrm{using}\:\mathrm{the}\:\mathrm{taylor}\:\mathrm{series}\:\mathrm{expansion}\:\mathrm{for}\:\mathrm{sin}\:\left({x}\right)\:\mathrm{arcsin}\:\left({x}\right)\:\mathrm{centred}\:\mathrm{at}\:\mathrm{0} \\ $$$$\:\mathrm{sin}\:\left({x}\right)\:\mathrm{arcsin}\:\left({x}\right)\:=\:{x}^{\mathrm{2}} \:+\:\frac{{x}^{\mathrm{6}} }{\mathrm{18}}\:+\:\frac{{x}^{\mathrm{8}} }{\mathrm{30}}\:+\:\frac{\mathrm{2669}\:{x}^{\mathrm{10}} }{\mathrm{113400}}\:+\:\frac{\mathrm{601}\:{x}^{\mathrm{12}} }{\mathrm{34020}}\:+\:... \\ $$$$\Rightarrow\:\int\:\mathrm{sin}\:\left({x}\right)\:\mathrm{arcsin}\:\left({x}\right)\:{dx}\:=\:\int\left({x}^{\mathrm{2}} \:+\:\frac{{x}^{\mathrm{6}} }{\mathrm{18}}\:+\:\frac{{x}^{\mathrm{8}} }{\mathrm{30}}\:+\:\frac{\mathrm{2669}\:{x}^{\mathrm{10}} }{\mathrm{113400}}\:+\:\frac{\mathrm{601}\:{x}^{\mathrm{12}} }{\mathrm{34020}}+...\right){dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{{x}^{\mathrm{3}} }{\mathrm{3}}\:+\:\frac{{x}^{\mathrm{7}} }{\mathrm{126}}\:+\:\frac{{x}^{\mathrm{9}} }{\mathrm{270}}\:+\:\frac{\mathrm{2669}\:{x}^{\mathrm{11}} }{\mathrm{1247400}}\:+\:\frac{\mathrm{601}\:{x}^{\mathrm{13}} }{\mathrm{442260}}\:+\:...\:+\:{k} \\ $$$$\mathrm{i}\:\mathrm{still}\:\mathrm{personally}\:\mathrm{think}\:\mathrm{this}\:\mathrm{problem}\:\mathrm{will}\:\mathrm{be}\:\mathrm{cool}\:\mathrm{if}\:\mathrm{we}\:\mathrm{had}\:\mathrm{boundary}\: \\ $$$$\mathrm{conditions}\:\:\mathrm{like}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{sin}\:\left({x}\right)\:\mathrm{arcsin}\:\left({x}\right)\:{dx}. \\ $$

Commented by M±th+et£s last updated on 30/Mar/20

thank you sir

$${thank}\:{you}\:{sir} \\ $$

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