Question Number 202125 by Calculusboy last updated on 21/Dec/23 $$\int\:\frac{\boldsymbol{{sin}}\left(\mathrm{3}\boldsymbol{{x}}\right)}{\mathrm{1}+\boldsymbol{{sin}}^{\mathrm{3}} \boldsymbol{{x}}}\boldsymbol{{dx}} \\ $$ Answered by Frix last updated on 21/Dec/23 $$\mathrm{Let}\:{s}=\mathrm{sin}\:{x} \\ $$$$\frac{\mathrm{sin}\:\mathrm{3}{x}}{\mathrm{1}+\mathrm{sin}^{\mathrm{3}} \:{x}}=\frac{{s}\left(\mathrm{4}{s}^{\mathrm{2}} −\mathrm{3}\right)}{\left({s}+\mathrm{1}\right)\left({s}^{\mathrm{2}}…
Question Number 202127 by Calculusboy last updated on 21/Dec/23 Answered by qaz last updated on 21/Dec/23 $$\int_{\pi/\mathrm{6}} ^{\pi/\mathrm{3}} \frac{{dx}}{\mathrm{1}+\mathrm{tan}^{\pi} \:{x}}=\int_{\pi/\mathrm{6}} ^{\pi/\mathrm{3}} \frac{{dx}}{\mathrm{1}+\mathrm{cot}\:^{\pi} {x}}=\frac{\mathrm{1}}{\mathrm{2}}\int_{\pi/\mathrm{6}} ^{\pi/\mathrm{3}} \left(\frac{\mathrm{1}}{\mathrm{1}+\mathrm{tan}^{\pi}…
Question Number 202120 by MATHEMATICSAM last updated on 21/Dec/23 $$\mathrm{If}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$${ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c}\:=\:\mathrm{0}.\:\mathrm{If}\:\:\frac{\mathrm{1}\:−\:\alpha}{\alpha}\:\mathrm{and}\:\frac{\mathrm{1}\:−\:\beta}{\beta}\:\mathrm{are} \\ $$$$\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{a}_{\mathrm{1}} {x}^{\mathrm{2}} \:+\:{b}_{\mathrm{1}} {x}\:+\:{c}_{\mathrm{1}} \:=\:\mathrm{0}.\:\mathrm{If} \\ $$$$\frac{{b}_{\mathrm{1}} }{{a}_{\mathrm{1}} }\:=\:{k}\:+\:\frac{{b}}{{c}}\:\mathrm{then}\:{k}\:=\:? \\ $$…
Question Number 202123 by BaliramKumar last updated on 21/Dec/23 $$\frac{\mathrm{1}}{\mathrm{1}×\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{3}×\mathrm{5}}\:+\:\frac{\mathrm{1}}{\mathrm{5}×\mathrm{7}}\:+\:……………\infty\:=\:? \\ $$ Answered by Rasheed.Sindhi last updated on 21/Dec/23 $${t}_{{n}} =\frac{\mathrm{1}}{\left(\mathrm{2}{n}−\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)} \\ $$$$\:\:\:\:\:{let}\:\:\frac{\mathrm{1}}{\left(\mathrm{2}{n}−\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)}=\frac{{a}}{\mathrm{2}{n}−\mathrm{1}}+\frac{{b}}{\mathrm{2}{n}+\mathrm{1}} \\ $$$$\:\:\:{a}\left(\mathrm{2}{n}+\mathrm{1}\right)+{b}\left(\mathrm{2}{n}−\mathrm{1}\right)=\mathrm{1}…
Question Number 202116 by Calculusboy last updated on 21/Dec/23 $$\sqrt{\mathrm{3}^{\boldsymbol{{x}}} }\:+\mathrm{1}=\mathrm{2}^{\boldsymbol{{x}}} \:\:\:\boldsymbol{{find}}\:\boldsymbol{{x}} \\ $$ Answered by Frix last updated on 21/Dec/23 $$\mathrm{Obviously}\:{x}=\mathrm{2} \\ $$$$\sqrt{\mathrm{3}^{\mathrm{2}} }+\mathrm{1}=\sqrt{\mathrm{9}}+\mathrm{1}=\mathrm{3}+\mathrm{1}=\mathrm{4}=\mathrm{2}^{\mathrm{2}}…
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Question Number 202146 by hardmath last updated on 21/Dec/23 $$ \\ $$There are three children in a family. Two of the children have blood group…
Question Number 202114 by MATHEMATICSAM last updated on 21/Dec/23 $$\mathrm{If}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$${ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c}\:=\:\mathrm{0}\:\mathrm{and}\:\alpha\:+\:{k}\:\mathrm{and}\:\beta\:+\:{k}\:\:\mathrm{are}\: \\ $$$$\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{lx}^{\mathrm{2}} \:+\:{mx}\:+\:{n}\:=\:\mathrm{0}\:\mathrm{then}\:\mathrm{prove} \\ $$$$\mathrm{that}\:{k}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{{b}}{{a}}\:−\:\frac{{m}}{{l}}\right). \\ $$ Answered by aleks041103 last updated…
Question Number 202109 by hardmath last updated on 20/Dec/23
Question Number 202104 by necx122 last updated on 20/Dec/23 Commented by necx122 last updated on 20/Dec/23 $${please}\:{help}\:{with}\:{this} \\ $$ Commented by cortano12 last updated on…