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Question Number 64270 by aliesam last updated on 16/Jul/19

∫((5sin(x) cos(x))/((cos(x)+1))^(1/3) ) dx

$$\int\frac{\mathrm{5}{sin}\left({x}\right)\:{cos}\left({x}\right)}{\sqrt[{\mathrm{3}}]{{cos}\left({x}\right)+\mathrm{1}}}\:{dx} \\ $$

Answered by Tanmay chaudhury last updated on 16/Jul/19

t^3 =1+cosx 3t^2 dt=−sinxdx ∫((5×(t^3 −1)×−3t^2 dt)/t) −15∫(t^4 −t)dt =(−15)×{(t^5 /5)−(t^2 /2)}+c =−3t^5 +((15)/2)t^2 +c =−3(1+cosx)^(5/3) +((15)/2)(1+cosx)^(2/3) +c

$${t}^{\mathrm{3}} =\mathrm{1}+{cosx}\:\:\mathrm{3}{t}^{\mathrm{2}} {dt}=−{sinxdx} \\ $$$$\int\frac{\mathrm{5}×\left({t}^{\mathrm{3}} −\mathrm{1}\right)×−\mathrm{3}{t}^{\mathrm{2}} {dt}}{{t}} \\ $$$$−\mathrm{15}\int\left({t}^{\mathrm{4}} −{t}\right){dt} \\ $$$$=\left(−\mathrm{15}\right)×\left\{\frac{{t}^{\mathrm{5}} }{\mathrm{5}}−\frac{{t}^{\mathrm{2}} }{\mathrm{2}}\right\}+{c} \\ $$$$=−\mathrm{3}{t}^{\mathrm{5}} +\frac{\mathrm{15}}{\mathrm{2}}{t}^{\mathrm{2}} +{c} \\ $$$$=−\mathrm{3}\left(\mathrm{1}+{cosx}\right)^{\frac{\mathrm{5}}{\mathrm{3}}} +\frac{\mathrm{15}}{\mathrm{2}}\left(\mathrm{1}+{cosx}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} +{c} \\ $$

Commented by Prithwish sen last updated on 16/Jul/19

Tanmay sir. Is it you ? Happy to see you sir. please don′t leave.

$$\mathrm{Tanmay}\:\mathrm{sir}.\:\mathrm{Is}\:\mathrm{it}\:\mathrm{you}\:?\:\mathrm{Happy}\:\mathrm{to}\:\mathrm{see}\:\mathrm{you}\:\mathrm{sir}. \\ $$$$\mathrm{please}\:\mathrm{don}'\mathrm{t}\:\mathrm{leave}. \\ $$

Commented by Tawa1 last updated on 16/Jul/19

Sir tanmay, were have you been. Hope you are good

$$\mathrm{Sir}\:\mathrm{tanmay},\:\mathrm{were}\:\mathrm{have}\:\mathrm{you}\:\mathrm{been}.\:\mathrm{Hope}\:\mathrm{you}\:\mathrm{are}\:\mathrm{good} \\ $$

Commented by aliesam last updated on 16/Jul/19

i am so happy to see you again sir and thank you for the sol

$${i}\:{am}\:{so}\:{happy}\:{to}\:{see}\:{you}\:{again}\:{sir}\:{and}\:{thank}\:{you}\:{for}\:{the}\:{sol} \\ $$

Commented by Tanmay chaudhury last updated on 16/Jul/19

this platform provide food of brain so i can not stay starved

$${this}\:{platform}\:{provide}\:{food}\:{of}\:{brain}\:{so}\:{i}\:{can}\:{not} \\ $$$${stay}\:{starved} \\ $$

Commented by Tawa1 last updated on 16/Jul/19

Hahahaha. welcome sir

$$\mathrm{Hahahaha}.\:\mathrm{welcome}\:\mathrm{sir} \\ $$

Commented by Prithwish sen last updated on 16/Jul/19

Sir are you from kolkata ?

$$\mathrm{Sir}\:\mathrm{are}\:\mathrm{you}\:\mathrm{from}\:\mathrm{kolkata}\:? \\ $$

Commented by Tanmay chaudhury last updated on 16/Jul/19

ha ami barrakcpore a thaki

$${ha}\:{ami}\:{barrakcpore}\:{a}\:{thaki} \\ $$

Commented by Prithwish sen last updated on 16/Jul/19

ami tribeni.kono din dekha hobe. barrackpure amar mamar bari

$$\mathrm{ami}\:\mathrm{tribeni}.\mathrm{kono}\:\mathrm{din}\:\mathrm{dekha}\:\mathrm{hobe}.\:\mathrm{barrackpure}\: \\ $$$$\mathrm{amar}\:\mathrm{mamar}\:\mathrm{bari} \\ $$

Commented by Tanmay chaudhury last updated on 16/Jul/19

achha thik ache..bhalo thakun..

$${achha}\:{thik}\:{ache}..{bhalo}\:{thakun}.. \\ $$

Commented by Prithwish sen last updated on 16/Jul/19

apnio.

$$\mathrm{apnio}. \\ $$

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