dx-sinx-cosx-1-

Question Number 160432 by mkam last updated on 29/Nov/21

$$\int\:\frac{{dx}}{{sinx}+{cosx}+\mathrm{1}} \\ $$

Answered by Ar Brandon last updated on 29/Nov/21

I=∫(dx/(sinx+cosx+1)) =∫(dx/(2sin((x/2))cos((x/2))+2cos^2 (x/2))) =(1/2)∫((sec^2 ((x/2)))/(tan((x/2))+1))dx=∫((d(tan(x/2)))/(tan(x/2)+1)) =ln∣tan(x/2)+1∣+C

$${I}=\int\frac{{dx}}{\mathrm{sin}{x}+\mathrm{cos}{x}+\mathrm{1}} \\ $$$$\:\:\:=\int\frac{{dx}}{\mathrm{2sin}\left(\frac{{x}}{\mathrm{2}}\right)\mathrm{cos}\left(\frac{{x}}{\mathrm{2}}\right)+\mathrm{2cos}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}} \\ $$$$\:\:\:=\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{sec}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)}{\mathrm{tan}\left(\frac{{x}}{\mathrm{2}}\right)+\mathrm{1}}{dx}=\int\frac{{d}\left(\mathrm{tan}\frac{{x}}{\mathrm{2}}\right)}{\mathrm{tan}\frac{{x}}{\mathrm{2}}+\mathrm{1}} \\ $$$$\:\:\:=\mathrm{ln}\mid\mathrm{tan}\frac{{x}}{\mathrm{2}}+\mathrm{1}\mid+{C} \\ $$

Commented by MJS_new last updated on 29/Nov/21

I had solved it, too. but we only get “?????” from this member if we don′t solve his questions fast enough so I decided to delete my solution.

$$\mathrm{I}\:\mathrm{had}\:\mathrm{solved}\:\mathrm{it},\:\mathrm{too}. \\ $$$$\mathrm{but}\:\mathrm{we}\:\mathrm{only}\:\mathrm{get}\:“?????''\:\mathrm{from}\:\mathrm{this}\:\mathrm{member} \\ $$$$\mathrm{if}\:\mathrm{we}\:\mathrm{don}'\mathrm{t}\:\mathrm{solve}\:\mathrm{his}\:\mathrm{questions}\:\mathrm{fast}\:\mathrm{enough} \\ $$$$\mathrm{so}\:\mathrm{I}\:\mathrm{decided}\:\mathrm{to}\:\mathrm{delete}\:\mathrm{my}\:\mathrm{solution}. \\ $$

Commented by Ar Brandon last updated on 29/Nov/21

Well... what can we do, Sir? Ignoring and reacting when we feel like doing so, in my opinion is the best way. Btw I don′t feel pressurized by any “???”.

$$\mathrm{Well}…\:\mathrm{what}\:\mathrm{can}\:\mathrm{we}\:\mathrm{do},\:\mathrm{Sir}?\:\mathrm{Ignoring} \\ $$$$\mathrm{and}\:\mathrm{reacting}\:\mathrm{when}\:\mathrm{we}\:\mathrm{feel}\:\mathrm{like}\:\mathrm{doing}\:\mathrm{so}, \\ $$$$\mathrm{in}\:\mathrm{my}\:\mathrm{opinion}\:\mathrm{is}\:\mathrm{the}\:\mathrm{best}\:\mathrm{way}.\:\mathrm{Btw}\:\mathrm{I}\:\mathrm{don}'\mathrm{t} \\ $$$$\mathrm{feel}\:\mathrm{pressurized}\:\mathrm{by}\:\mathrm{any}\:“???''.\: \\ $$

Commented by MJS_new last updated on 29/Nov/21

me neither. I just decided not to answer any more.

$$\mathrm{me}\:\mathrm{neither}.\:\mathrm{I}\:\mathrm{just}\:\mathrm{decided}\:\mathrm{not}\:\mathrm{to}\:\mathrm{answer}\:\mathrm{any} \\ $$$$\mathrm{more}. \\ $$

Commented by mr W last updated on 30/Nov/21

$$“????''\:{means}\:{nothing}\:{but}\:“{help}!\:{help}!, \\ $$$${be}\:{quick}!\:{why}\:{still}\:{no}\:{help}?''.\:{if}\:{it}\:{is} \\ $$$${only}\:{this},\:{it}'{s}\:{still}\:{ok}\:{to}\:{me}.\:{but}\:{the} \\ $$$${problem}\:{is}\:{when}\:{you}\:{really}\:{help},\:{no} \\ $$$${response}\:{comes}\:{back}.\:{a}\:{man}\:{can}\:{cry} \\ $$$${for}\:{help},\:{but}\:{is}\:{not}\:{able}\:{to}\:{say}\:{thanks} \\ $$$${to}\:{the}\:{helpers}. \\ $$

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