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Question Number 42358 by Tawa1 last updated on 24/Aug/18

∫_( −1) ^( 1) (x^(2015) /(((1 + x))^(1/(2015)) + ((1 − x))^(1/(2015)) )) dx

$$\int_{\:−\mathrm{1}} ^{\:\mathrm{1}} \:\frac{\mathrm{x}^{\mathrm{2015}} }{\sqrt[{\mathrm{2015}}]{\mathrm{1}\:+\:\mathrm{x}}\:\:+\:\:\sqrt[{\mathrm{2015}}]{\mathrm{1}\:−\:\mathrm{x}}\:}\:\:\mathrm{dx} \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 24/Aug/18

Υ(x)=(x^(2015) /((1+x)^(1/(2015)) +(1−x)^(1/(2015)) )) Υ(−x)=(((−x)^(2015) )/((1−x)^(1/(2015)) +(1+x)^(1/(2015)) ))=−Υ(x) so f(x) is odd function... ∫_a ^b f(x)dx=∫_a ^b f(a+b−x)dx formula... ∫_(−1) ^1 Υ(x)dx=∫_(−1) ^1 Υ(1−1−x)dx =∫_(−1) ^1 −Υ(x)dx so 2∫_(−1) ^1 Υ(x)dx=0 ∫_(−1) ^1 Υ(x)dx=0 hence ∫_(−1) ^1 (x^(2015) /((1+x)^(1/(2015)) +(1−x)^(1/(2015)) ))dx =0

$$\Upsilon\left({x}\right)=\frac{{x}^{\mathrm{2015}} }{\left(\mathrm{1}+{x}\right)^{\frac{\mathrm{1}}{\mathrm{2015}}} +\left(\mathrm{1}−{x}\right)^{\frac{\mathrm{1}}{\mathrm{2015}}} } \\ $$$$\Upsilon\left(−{x}\right)=\frac{\left(−{x}\right)^{\mathrm{2015}} }{\left(\mathrm{1}−{x}\right)^{\frac{\mathrm{1}}{\mathrm{2015}}} +\left(\mathrm{1}+{x}\right)^{\frac{\mathrm{1}}{\mathrm{2015}}} }=−\Upsilon\left({x}\right) \\ $$$${so}\:{f}\left({x}\right)\:{is}\:{odd}\:{function}... \\ $$$$\int_{{a}} ^{{b}} {f}\left({x}\right){dx}=\int_{{a}} ^{{b}} {f}\left({a}+{b}−{x}\right){dx}\:{formula}... \\ $$$$\int_{−\mathrm{1}} ^{\mathrm{1}} \Upsilon\left({x}\right){dx}=\int_{−\mathrm{1}} ^{\mathrm{1}} \Upsilon\left(\mathrm{1}−\mathrm{1}−{x}\right){dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\int_{−\mathrm{1}} ^{\mathrm{1}} −\Upsilon\left({x}\right){dx} \\ $$$${so}\:\mathrm{2}\int_{−\mathrm{1}} ^{\mathrm{1}} \Upsilon\left({x}\right){dx}=\mathrm{0}\:\:\:\:\int_{−\mathrm{1}} ^{\mathrm{1}} \Upsilon\left({x}\right){dx}=\mathrm{0} \\ $$$${hence}\:\int_{−\mathrm{1}} ^{\mathrm{1}} \:\frac{{x}^{\mathrm{2015}} }{\left(\mathrm{1}+{x}\right)^{\frac{\mathrm{1}}{\mathrm{2015}}} +\left(\mathrm{1}−{x}\right)^{\frac{\mathrm{1}}{\mathrm{2015}}} }{dx}\:\:\:\:=\mathrm{0} \\ $$

Commented by Tawa1 last updated on 24/Aug/18

God bless you sir. what is this sign (Υ) means ??

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}.\:\:\mathrm{what}\:\mathrm{is}\:\mathrm{this}\:\mathrm{sign}\:\left(\Upsilon\right)\:\mathrm{means}\:?? \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 24/Aug/18

its a symbol...but who red marked me and why

$${its}\:{a}\:{symbol}...{but}\:{who}\:{red}\:{marked}\:{me}\:{and}\:{why} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Commented by Tawa1 last updated on 24/Aug/18

Owk. Just thinking if the symbol means something. Maybe the person that redmark you want to like your solution but mistakenly redmark. God bless you sir.

$$\mathrm{Owk}.\:\mathrm{Just}\:\mathrm{thinking}\:\mathrm{if}\:\mathrm{the}\:\mathrm{symbol}\:\mathrm{means}\:\mathrm{something}. \\ $$$$\mathrm{Maybe}\:\mathrm{the}\:\mathrm{person}\:\mathrm{that}\:\mathrm{redmark}\:\mathrm{you}\:\mathrm{want}\:\mathrm{to}\:\mathrm{like}\:\mathrm{your}\:\mathrm{solution} \\ $$$$\mathrm{but}\:\mathrm{mistakenly}\:\mathrm{redmark}. \\ $$$$\:\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}.\: \\ $$

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