Question Number 207129 by hardmath last updated on 07/May/24 $$\mathrm{Find}:\:\:\:\:\:\int\:\frac{\mathrm{x}\:+\:\mathrm{4}}{\mathrm{x}\:+\:\mathrm{3}}\:\mathrm{dx}\:=\:? \\ $$ Answered by mathzup last updated on 08/May/24 $$\int\frac{{x}+\mathrm{4}}{{x}+\mathrm{3}}{dx}=\int\frac{{x}+\mathrm{3}+\mathrm{1}}{{x}+\mathrm{3}}{dx}\:=\int{dx}\:+\int\frac{{dx}}{{x}+\mathrm{3}} \\ $$$$={x}+{ln}\mid{x}+\mathrm{3}\mid\:+{c} \\ $$ Terms…
Question Number 207156 by hardmath last updated on 07/May/24 $$\mathrm{Find}:\:\:\:\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{10}°}\:−\:\frac{\sqrt{\mathrm{3}}}{\mathrm{cos}\:\mathrm{10}°}\:\:=\:\:? \\ $$ Answered by A5T last updated on 07/May/24 $$=\frac{\mathrm{2}{sin}\mathrm{30}}{{sin}\mathrm{10}}−\frac{\mathrm{2}{cos}\mathrm{30}}{{cos}\mathrm{10}}=\frac{\mathrm{2}\left[{sin}\mathrm{30}{cos}\mathrm{10}−{cos}\mathrm{30}{sin}\mathrm{10}\right]}{{sin}\mathrm{10}{cos}\mathrm{10}} \\ $$$$=\frac{\mathrm{2}\left[{sin}\left(\mathrm{30}−\mathrm{10}\right)\right]}{{sin}\mathrm{10}{cos}\mathrm{10}}=\frac{\mathrm{4}{sin}\mathrm{20}}{\mathrm{2}{sin}\mathrm{10}{cos}\mathrm{10}={sin}\mathrm{20}}=\mathrm{4} \\ $$ Terms…
Question Number 207130 by hardmath last updated on 07/May/24 $$\mathrm{Find}:\:\:\:\int\:\frac{\mathrm{x}\:−\:\mathrm{1}}{\mathrm{x}\:−\:\mathrm{2}}\:\mathrm{dx}\:\:=\:\:? \\ $$ Answered by Frix last updated on 07/May/24 $$\int\frac{{x}\pm{a}}{{x}\pm{b}}{dx}=\int{dx}+\left(\pm{a}\mp{b}\right)\int\frac{{dx}}{{x}\pm{b}}= \\ $$$$={x}+\left(\pm{a}\mp{b}\right)\mathrm{ln}\:\mid{x}\pm{b}\mid\:+{C} \\ $$ Commented…
Question Number 207157 by hardmath last updated on 07/May/24 $$\mathrm{Find}:\:\:\:\frac{\mathrm{5}\:\mathrm{sin}\:\frac{\pi}{\mathrm{6}}}{\frac{\mathrm{1}}{\mathrm{tg}\:\mathrm{75}°}\:\:−\:\:\mathrm{tg}\:\mathrm{75}°}\:\:=\:\:? \\ $$ Answered by A5T last updated on 07/May/24 $$\frac{\mathrm{1}}{{tan}\mathrm{75}}−{tan}\mathrm{75}=\frac{{cos}\mathrm{75}}{{sin}\mathrm{75}}−\frac{{sin}\mathrm{75}}{{cos}\mathrm{75}}=\frac{{cos}^{\mathrm{2}} \mathrm{75}−{sin}^{\mathrm{2}} \mathrm{75}}{{sin}\mathrm{75}{cos}\mathrm{75}} \\ $$$$=\frac{\mathrm{2}{cos}\mathrm{150}}{{sin}\mathrm{150}}=\frac{\mathrm{2}\left(\frac{−\sqrt{\mathrm{3}}}{\mathrm{2}}\right)}{\frac{\mathrm{1}}{\mathrm{2}}}=−\mathrm{2}\sqrt{\mathrm{3}} \\…
Question Number 207127 by hardmath last updated on 07/May/24 $$\mathrm{If}\:\:\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{sin}^{\mathrm{4}} \:\mathrm{2x} \\ $$$$\mathrm{Find}\:\:\:\:\:\mathrm{f}\:^{'} \:\left(\frac{\pi}{\mathrm{12}}\right)\:=\:? \\ $$ Answered by Skabetix last updated on 07/May/24 $$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{sin}^{\mathrm{4}} \left(\mathrm{2x}\right)…
Question Number 207116 by hardmath last updated on 06/May/24 $$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:=\:\mathrm{x}\:−\:\mathrm{2y}}\\{\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{y}\:−\:\mathrm{2x}}\end{cases}\:\:\:\:\:\Rightarrow\:\:\:\:\:\mathrm{x}\:−\:\mathrm{y}\:=\:? \\ $$ Answered by A5T last updated on 06/May/24 $${x}^{\mathrm{2}} −{y}^{\mathrm{2}} ={x}−{y}+\mathrm{2}\left({x}−{y}\right)=\mathrm{3}\left({x}−{y}\right) \\…
Question Number 207092 by hardmath last updated on 06/May/24 $$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:=\:\mathrm{x}\:−\:\mathrm{6y}}\\{\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{y}\:−\:\mathrm{6x}}\end{cases}\:\:\:\:\:\Rightarrow\:\:\:\:\mathrm{x}\:+\:\mathrm{y}\:=\:? \\ $$ Answered by A5T last updated on 06/May/24 $${x}^{\mathrm{2}} −{y}^{\mathrm{2}} ={x}−{y}−\mathrm{6}{y}+\mathrm{6}{x}=\mathrm{7}{x}−\mathrm{7}{y} \\…
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Question Number 207043 by necx122 last updated on 04/May/24 $${Solve}\:{for}\:{the}\:{domain}\:{and}\:{range}\:{of}\:{f}\left({x}\right)\:{if} \\ $$$${f}\left({x}\right)\:=\:\sqrt{\frac{\mathrm{3}{x}−\mathrm{5}}{{x}+\mathrm{3}}} \\ $$ Answered by Frix last updated on 04/May/24 $$\mathrm{1}.\:{x}+\mathrm{3}\neq\mathrm{0} \\ $$$${x}\neq−\mathrm{3} \\…
Question Number 206997 by efronzo1 last updated on 03/May/24 $$\mathrm{Given}\:\mathrm{a}_{\mathrm{n}+\mathrm{1}} =\:\mathrm{3}−\mathrm{6a}_{\mathrm{n}} \:\mathrm{and}\:\mathrm{a}_{\mathrm{4}} =−\mathrm{9} \\ $$$$\:\mathrm{Find}\:\mathrm{a}_{\mathrm{n}} =?\: \\ $$ Answered by mr W last updated on…