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

∫ (dx/(sinx+cosx+1))

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

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.

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

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