Question Number 15797 by tawa tawa last updated on 14/Jun/17 $$\mathrm{If}\:\:\left(\mathrm{a}\:+\:\mathrm{bx}\right)\mathrm{e}^{\mathrm{y}/\mathrm{x}} \:=\:\mathrm{x},\:\:\:\mathrm{where}\:\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{are}\:\mathrm{constant},\: \\ $$$$\mathrm{prove}\:\mathrm{that},:\:\:\:\mathrm{x}^{\mathrm{3}} \mathrm{y}''\:=\:\left(\mathrm{xy}'\:−\:\mathrm{y}\right)^{\mathrm{2}} \\ $$ Commented by tawa tawa last updated on 14/Jun/17…
Question Number 15741 by ajfour last updated on 13/Jun/17 $${Find}\:{the}\:{minimum}\:{value}\:\:{of} \\ $$$$\:\:\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} ,\:{with}\:{the}\:{condition} \\ $$$$\:{ax}+{by}+{cz}={p}\:. \\ $$ Answered by mrW1 last updated on…
Question Number 81158 by ~blr237~ last updated on 09/Feb/20 $${S}=\frac{\mathrm{1}}{{cos}\mathrm{1}}+\frac{\mathrm{1}}{{cos}\mathrm{1}{cos}\mathrm{2}}+……+\frac{\mathrm{1}}{{cos}\mathrm{87}{cos}\mathrm{88}} \\ $$$${K}={tan}\mathrm{1}{tan}\mathrm{2}+{tan}\mathrm{3}{tan}\mathrm{4}+……+{tan}\mathrm{87}{tan}\mathrm{88} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 146632 by KINMATICS last updated on 14/Jul/21 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 146602 by mnjuly1970 last updated on 14/Jul/21 Answered by qaz last updated on 15/Jul/21 $$\int\frac{\mathrm{1}−\mathrm{cos}\:\left(\mathrm{12x}\right)}{\mathrm{2sin}\:\mathrm{x}}\mathrm{dx}=\int\frac{\mathrm{sin}\:^{\mathrm{2}} \left(\mathrm{6x}\right)}{\mathrm{sin}\:\mathrm{x}}\mathrm{dx} \\ $$$$=\mathrm{4}\int\frac{\left[\mathrm{sin}\:\left(\mathrm{3x}\right)\mathrm{cos}\:\left(\mathrm{3x}\right)\right]^{\mathrm{2}} }{\mathrm{sin}\:\mathrm{x}}\mathrm{dx} \\ $$$$=\mathrm{4}\int\frac{\left[\left(\mathrm{3sin}\:\mathrm{x}−\mathrm{4sin}\:^{\mathrm{3}} \mathrm{x}\right)\left(\mathrm{4cos}\:^{\mathrm{3}} \mathrm{x}−\mathrm{3cos}\:\mathrm{x}\right)\right]^{\mathrm{2}}…
Question Number 146523 by mnjuly1970 last updated on 13/Jul/21 Answered by ajfour last updated on 13/Jul/21 $${f}\left(\frac{{x}+{y}}{\mathrm{2}}\right)=\frac{\mathrm{2}+{f}\left({x}\right)+{f}\left({y}\right)}{\mathrm{3}} \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{2} \\ $$$${f}\left({x}\right)=\mathrm{2}{x}+\mathrm{2} \\ $$$$\mathrm{3}{f}\left({x}+{y}\right)=\mathrm{2}+{f}\left(\mathrm{2}{x}\right)+{f}\left(\mathrm{2}{y}\right) \\ $$$$\Rightarrow…
Question Number 146442 by qaz last updated on 13/Jul/21 $$\frac{\mathrm{d}}{\mathrm{dn}}\mid_{\mathrm{n}=\mathrm{1}} \mathrm{H}_{\mathrm{n}} =? \\ $$ Answered by mnjuly1970 last updated on 13/Jul/21 $$=\:\frac{{d}}{{dn}}\left(\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{1}−{x}^{\:{n}} }{\mathrm{1}−{x}}\right)=\int_{\mathrm{0}}…
Question Number 80863 by ~blr237~ last updated on 07/Feb/20 $$\:{Let}\:{W}\:{the}\:{lambert}\:{function}\:{defined}\:{as}\:{W}\left({xe}^{{x}} \right)={x}\:\:\:{x}\geqslant\mathrm{0} \\ $$$${Prove}\:{that}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\:{W}\left(−{ulnu}\right)}{{u}}{du}=\frac{\zeta\left(\mathrm{2}\right)}{\mathrm{2}}\:\: \\ $$ Answered by Kamel Kamel last updated on 08/Feb/20…
Question Number 146366 by liberty last updated on 13/Jul/21 Answered by iloveisrael last updated on 13/Jul/21 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 80816 by ~blr237~ last updated on 06/Feb/20 $${let}\:\:\mathrm{0}<{a}<{b}\:\:{prove}\:{that} \\ $$$$\:{ln}\left(\mathrm{1}+\frac{{a}}{{b}}\right){ln}\left(\mathrm{1}+\frac{{b}}{{a}}\right)<\:\left({ln}\mathrm{2}\right)^{\mathrm{2}} \:\: \\ $$ Commented by ~blr237~ last updated on 06/Feb/20 $${Sir}\:\:,\:{i}\:{look}\:{like}\:\:{at}\:\left(\mathrm{3}−\mathrm{4}\right)\:{crossing}\:{you}\:{uze} \\ $$$${A}\leqslant{B}\:{and}\:{B}\geqslant{C}\:\Rightarrow\:{A}\leqslant{C}\:\:\:\:??\:…