Question Number 210126 by mnjuly1970 last updated on 31/Jul/24 Answered by Frix last updated on 31/Jul/24 $$\mathrm{These}\:\mathrm{substitutions}\:\mathrm{make}\:\mathrm{it}\:\mathrm{easy}\:\mathrm{to}\:\mathrm{see} \\ $$$$\mathrm{what}'\mathrm{s}\:\mathrm{going}\:\mathrm{on}: \\ $$$$ \\ $$$$\mathrm{Let}\:{x}=\mathrm{sin}\:\alpha\:\overset{\left[\mathrm{differentiate}\right]} {\Rightarrow} \\…
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Question Number 210127 by essaad last updated on 31/Jul/24 Answered by lepuissantcedricjunior last updated on 01/Aug/24 $$\int_{\mathrm{1}} ^{\mathrm{2}} \frac{\boldsymbol{{ln}}\left(\mathrm{1}+\boldsymbol{{x}}\right)−\boldsymbol{{lnx}}}{\boldsymbol{{x}}^{\mathrm{2}} }\boldsymbol{{dx}}=\int_{\mathrm{1}} ^{\mathrm{2}} \frac{\boldsymbol{{ln}}\left(\mathrm{1}+\boldsymbol{{x}}\right)}{\boldsymbol{{x}}^{\mathrm{2}} }\boldsymbol{{dx}}−\int_{\mathrm{0}} ^{\mathrm{2}} \frac{\boldsymbol{{lnx}}}{\boldsymbol{{x}}^{\mathrm{2}}…
Question Number 210091 by mr W last updated on 30/Jul/24 $${find}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}{n}^{\mathrm{2}} }\right)=? \\ $$ Commented by AlagaIbile last updated on 30/Jul/24 $$\:\:\underset{{n}=\mathrm{1}}…
Question Number 210079 by Spillover last updated on 30/Jul/24 $${Find}\:{directional}\:{derivatives}\left({D}_{{v}} \right){of}\:\: \\ $$$${f}\left({x},{y},{z}\right)=\mathrm{3}{xy}^{\mathrm{3}} −\mathrm{2}{xz}^{\mathrm{2}} \:\:{in}\:{the}\:{direction}\:{of}\:{the} \\ $$$${v}=\mathrm{2}{i}−\mathrm{3}{j}+\mathrm{6}{k}. \\ $$$${then}\:{Evaluate}\:{directional}\:{derivatives}\: \\ $$$${at}\:{the}\:{point}\:\left(\mathrm{3},\mathrm{1},−\mathrm{2}\right) \\ $$ Terms of…
Question Number 210078 by Spillover last updated on 29/Jul/24 $${Find}\:{the}\:{directional}\:{derivative}\:{of} \\ $$$${f}\left({x},{y}\right)=\mathrm{4}{x}^{\mathrm{3}} −\mathrm{3}{x}^{\mathrm{2}} {y}^{\mathrm{2}} \:\:\:{in}\:{the}\:{direction}\:{given} \\ $$$${by}\:{the}\:{angle}\:\theta=\frac{\pi}{\mathrm{3}}\: \\ $$$${and}\:{also}\:{Evaluate}\:{directional}\:{derivatives} \\ $$$${at}\:{the}\:{point}\:\left(\mathrm{1},\mathrm{2}\right) \\ $$ Answered by…
Question Number 210036 by peter frank last updated on 29/Jul/24 Answered by Prithwish last updated on 29/Jul/24 $${ab}=\left(\frac{\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} \theta}{\mathrm{cos}\:\theta}\right)\left(\frac{\mathrm{1}−\mathrm{sin}\:^{\mathrm{2}} \theta}{\mathrm{sin}\:\theta}\right) \\ $$$${ab}=\mathrm{sin}\:\theta\mathrm{cos}\:\theta \\ $$$${a}^{\mathrm{2}} +\overset{\mathrm{2}}…
Question Number 210072 by peter frank last updated on 29/Jul/24 Answered by Frix last updated on 29/Jul/24 $$\mathrm{The}\:\mathrm{incircle}\:\mathrm{of}\:\mathrm{a}\:\mathrm{rectangular}\:\mathrm{triangle}\:\mathrm{with} \\ $$$$\mathrm{sides}\:{a},\:{b},\:\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }\:\mathrm{is}\:\frac{{a}+{b}−\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }}{\mathrm{2}}\:\:\:\:\:\left(\ast\right) \\…
Question Number 210034 by peter frank last updated on 29/Jul/24 Commented by peter frank last updated on 29/Jul/24 $$\mathrm{solve}\:\mathrm{for}\:\mathrm{x} \\ $$ Commented by Frix last…
Question Number 210080 by Spillover last updated on 30/Jul/24 $${Given}\:{that}\:\:{det}\:\begin{bmatrix}{{a}}&{{b}}&{{c}}\\{{d}}&{{e}}&{{f}}\\{{g}}&{{h}}&{{i}}\end{bmatrix}={n} \\ $$$$ \\ $$$${find}\:{det}\begin{bmatrix}{{d}+\mathrm{2}{a}}&{{e}+\mathrm{2}{b}}&{{f}+\mathrm{2}{c}}\\{\mathrm{2}{a}}&{\mathrm{2}{b}}&{\mathrm{2}{c}}\\{\mathrm{4}{g}}&{\mathrm{4}{h}}&{\mathrm{4}{i}}\end{bmatrix} \\ $$$$ \\ $$ Commented by Frix last updated on 30/Jul/24…