Question Number 85694 by john santu last updated on 24/Mar/20 $$\mathrm{log}_{\left(\frac{{x}}{{x}−\mathrm{3}}\right)} \left(\mathrm{7}\right)\:<\:\mathrm{log}_{\left(\frac{{x}}{\mathrm{3}}\right)} \:\left(\mathrm{7}\right)\: \\ $$ Commented by jagoll last updated on 24/Mar/20 $$\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{7}} \left(\frac{\mathrm{x}}{\mathrm{x}−\mathrm{3}}\right)}\:<\:\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{7}} \left(\frac{\mathrm{x}}{\mathrm{3}}\right)}…
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Question Number 20157 by ajfour last updated on 23/Aug/17 Commented by ajfour last updated on 23/Aug/17 $${Compute}\:{the}\:{volume}\:{common}\:{to}\:{the} \\ $$$${sphere}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\mathrm{4}\:\:{and}\:\:{the}\:{inside}\:{of} \\ $$$${the}\:{paraboloid}\:\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}}…
Question Number 151230 by mnjuly1970 last updated on 19/Aug/21 $$ \\ $$$$\:\:\:\:{prove}: \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{ln}\:\left(\:\mathrm{1}+{x}^{\:\mathrm{2}} \right)}{{x}^{\:\mathrm{2}} \left(\mathrm{1}+{x}^{\:\mathrm{2}} \right)}{dx}=\:\pi\:{ln}\left(\frac{{e}}{\mathrm{2}}\:\right)\:.. \\ $$ Commented by Lordose last…
Question Number 20156 by tawa tawa last updated on 22/Aug/17 $$\mathrm{Solve}:\:\mathrm{inverse}\:\mathrm{laplace}.\:\:\:\:\mathrm{L}^{−\mathrm{1}} \left(\frac{\mathrm{s}}{\mathrm{s}^{\mathrm{2}\:} +\:\mathrm{6s}\:+\:\mathrm{25}}\right) \\ $$ Answered by ajfour last updated on 23/Aug/17 $$\mathscr{L}^{−\mathrm{1}} \left(\frac{{s}}{{s}^{\mathrm{2}} +\mathrm{6}{s}+\mathrm{25}}\right)=\mathscr{L}^{−\mathrm{1}}…
Question Number 151224 by mathdanisur last updated on 19/Aug/21 Answered by Rasheed.Sindhi last updated on 19/Aug/21 $${Let}\:{there}'{re}\:{x}\:{boys}\:{and}\:{y}\:{girls} \\ $$$${minimum}\:{number}\:{of}\:{slices}\:{eaten} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{4}{x}+\mathrm{2}{y} \\ $$$$\left(\mathrm{4}\:{slices}/{boy}\:\:\&\:\mathrm{2}\:{slices}/{girl}\right) \\ $$$$\mathrm{4}{x}+\mathrm{2}{y}>\mathrm{16}\:\:\left(\mathrm{2}\:{cakes},\:\mathrm{8}\:{slices}\:{per}\:{each}\:{cake}\right)…
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Question Number 151221 by peter frank last updated on 19/Aug/21 $$\int\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{sin}\:\left(\mathrm{x}−\mathrm{a}\right)}\mathrm{dx} \\ $$ Commented by puissant last updated on 19/Aug/21 $${Q}=\int\frac{{sinx}}{{sin}\left({x}−{a}\right)}{dx} \\ $$$$=\int\frac{{sin}\left({x}−{a}+{a}\right)}{{sin}\left({x}−{a}\right)}{dx}\:\: \\ $$$$=\int\frac{{sin}\left({x}−{a}\right){cosa}+{cos}\left({x}−{a}\right){sina}}{{sin}\left({x}−{a}\right)}{dx}…
Question Number 151220 by peter frank last updated on 19/Aug/21 $$\mathrm{show}\:\mathrm{that}\: \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{cos}\:\mathrm{x}+\mathrm{sin}\:\mathrm{x}+\mathrm{1}}\mathrm{dx}=\frac{\mathrm{1}}{\mathrm{4}}\left(\pi−\mathrm{2ln}\:\mathrm{2}\right) \\ $$ Answered by ArielVyny last updated on 19/Aug/21 $${t}={tan}\left(\frac{{x}}{\mathrm{2}}\right)\rightarrow{dt}=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+{t}^{\mathrm{2}}…
Question Number 20149 by ajfour last updated on 22/Aug/17 Commented by ajfour last updated on 22/Aug/17 $${The}\:{roof}\:{is}\:{even}\:{a}\:{flat}\:{plane}.{Lengths} \\ $$$$\boldsymbol{{a}},\:\boldsymbol{{b}},\:\boldsymbol{{c}},\:\boldsymbol{{y}}\:{are}\:{perpendicular}\:{to}\:{the} \\ $$$${bottom}\:{rectangular}\:{face}\:{with}\:{sides} \\ $$$$\boldsymbol{{h}}\:{and}\:\boldsymbol{{l}}\:.\:{Find}\:{height}\:\boldsymbol{{y}}\:{and}\:{the}\:\:{volume} \\ $$$${enclosed}\:{by}\:{the}\:{surfaces}.…