Question Number 221815 by alvan545 last updated on 11/Jun/25 Answered by mr W last updated on 11/Jun/25 $${say}\:{AC}={c} \\ $$$${BC}^{\mathrm{2}} ={CD}^{\mathrm{2}} \\ $$$${a}^{\mathrm{2}} +{c}^{\mathrm{2}} −\mathrm{2}{ac}\:\mathrm{cos}\:\theta={b}^{\mathrm{2}}…
Question Number 221843 by wewji12 last updated on 11/Jun/25 $$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\:{x}^{\mathrm{2}} \mathrm{csc}^{\mathrm{2}} \left({x}\right)\mathrm{d}{x} \\ $$$$\int\:\:−\frac{\mathrm{4}{z}^{\mathrm{2}} }{\left({e}^{\boldsymbol{{i}}{z}} −{e}^{−\boldsymbol{{i}}{z}} \right)^{\mathrm{2}} }\:\mathrm{d}{z}=\int\:\:\frac{{z}^{\mathrm{2}} {e}^{\mathrm{2}\boldsymbol{{i}}{z}} }{\left({e}^{\mathrm{2}\boldsymbol{{i}}{z}} −\mathrm{1}\right)^{\mathrm{2}} }\:\mathrm{d}{z} \\…
Question Number 221869 by fantastic last updated on 12/Jun/25 $${if}\:{a}^{\mathrm{3}−{x}} .{b}^{\mathrm{5}{x}} ={a}^{\mathrm{5}+{x}} .{b}^{\mathrm{3}{x}} \:{then}\:{show}\:{that} \\ $$$${x}\mathrm{log}\:\left(\frac{{b}}{{a}}\right)=\mathrm{log}\:{a} \\ $$ Answered by fantastic last updated on 12/Jun/25…
Question Number 221870 by fantastic last updated on 11/Jun/25 Answered by mehdee7396 last updated on 12/Jun/25 $${S}_{\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{2}}{AB}×{OM}\:\:\:\:\&\:\:\:{S}_{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{2}}{CD}×{ON} \\ $$$${AB}={CD}\:\:\Rightarrow{S}_{\mathrm{1}} +{S}_{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{2}}{AB}×\left({OM}+{ON}\right)=\frac{{AB}×{MN}}{\mathrm{2}}=\mathrm{16} \\ $$$$\Rightarrow{S}={AB}×{MN}=\mathrm{32}…
Question Number 221838 by Ghisom last updated on 11/Jun/25 $$\underset{\mathrm{0}} {\overset{\infty} {\int}}\left(\frac{\mathrm{arctan}\:{x}}{{x}}\right)^{\mathrm{2}} {dx}=? \\ $$ Answered by wewji12 last updated on 11/Jun/25 $$\mathrm{tan}^{−\mathrm{1}} \left({t}\right)=\rho\:\rightarrow\:\mathrm{d}{t}=\mathrm{sec}^{\mathrm{2}} \left(\rho\right)\mathrm{d}\rho…
Question Number 221832 by MrGaster last updated on 11/Jun/25 $${f}'\left({x}\right)=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{f}\left({x}+{h}\right)−{f}\left({x}\right)}{{h}} \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{c}−{c}}{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}0}=\mathrm{0} \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left({x}+{h}\right)^{\mathrm{2}} −{x}^{\mathrm{2}} }{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{2}{xh}+{h}^{\mathrm{2}} }{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\mathrm{2}{x}+{h}\right)=\mathrm{2}{x} \\ $$$$\underset{{h}\rightarrow\mathrm{0}}…
Question Number 221829 by fantastic last updated on 11/Jun/25 $$\sqrt{\mathrm{70}.\mathrm{71}.\mathrm{72}.\mathrm{73}+\mathrm{1}} \\ $$ Answered by aleks041103 last updated on 11/Jun/25 $$\mathrm{70}.\mathrm{71}.\mathrm{72}.\mathrm{73}+\mathrm{1}= \\ $$$$=\mathrm{70}.\mathrm{73}.\left(\mathrm{70}+\mathrm{1}\right)\left(\mathrm{73}−\mathrm{1}\right)+\mathrm{1}= \\ $$$$=\mathrm{70}.\mathrm{73}.\left(\mathrm{70}.\mathrm{73}+\mathrm{73}−\mathrm{70}−\mathrm{1}\right)+\mathrm{1}= \\…
Question Number 221830 by Nicholas666 last updated on 11/Jun/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\sqrt{{x}}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:+\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:{dx} \\ $$$$ \\ $$ Terms of Service Privacy Policy…
Question Number 221863 by fantastic last updated on 11/Jun/25 $${If}\:{a}\:{and}\:{b}\:{are}\:{whole}\:{numbers}\:{such}\:{a}^{{b}} =\mathrm{121} \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:\left({a}−\mathrm{1}\right)^{{b}+\mathrm{1}} \\ $$ Commented by Tawa11 last updated on 11/Jun/25 $$\mathrm{11}^{\mathrm{2}} \:\:=\:\:\mathrm{121} \\…
Question Number 221831 by Nicholas666 last updated on 11/Jun/25 $$\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\:\frac{\sqrt{{x}}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:+\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:{dx} \\ $$$$ \\ $$ Answered by MathematicalUser2357 last updated on 25/Jun/25…